1.7 multiplication ii w
TRANSCRIPT
![Page 2: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/2.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
![Page 3: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/3.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
In mathematics, we are also interested in the properties and relations.
![Page 4: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/4.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
In mathematics, we are also interested in the properties and relations.
Properties of Multiplication
![Page 5: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/5.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
In mathematics, we are also interested in the properties and relations.
Properties of MultiplicationWe note from before before that
* (0 * x = 0 * x = 0) The product of zero with any number is 0.
![Page 6: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/6.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
In mathematics, we are also interested in the properties and relations.
In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .
Properties of MultiplicationWe note from before before that
* (0 * x = 0 * x = 0) The product of zero with any number is 0.
![Page 7: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/7.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
In mathematics, we are also interested in the properties and relations.
In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .
Properties of MultiplicationWe note from before before that
* (0 * x = 0 * x = 0) The product of zero with any number is 0.
For example, A*B*0*C = 0 where A, B ,and C are numbers.
![Page 8: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/8.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
In mathematics, we are also interested in the properties and relations.
In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .
Properties of MultiplicationWe note from before before that
* (0 * x = 0 * x = 0) The product of zero with any number is 0.
* (1 * x = x * 1 = x) The product of 1 with any number x is x.
For example, A*B*0*C = 0 where A, B ,and C are numbers.
![Page 9: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/9.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
In mathematics, we are also interested in the properties and relations.
In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .
Properties of MultiplicationWe note from before before that
* (0 * x = 0 * x = 0) The product of zero with any number is 0.
* (1 * x = x * 1 = x) The product of 1 with any number x is x.In other words, 1 is the “preserver” in multiplication, It preserves anything multiplies with it .
For example, A*B*0*C = 0 where A, B ,and C are numbers.
![Page 10: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/10.jpg)
In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.
Multiplication II
In mathematics, we are also interested in the properties and relations.
In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .
Properties of MultiplicationWe note from before before that
For example, A*1*B*1*C = A*B*C.
* (0 * x = 0 * x = 0) The product of zero with any number is 0.
* (1 * x = x * 1 = x) The product of 1 with any number x is x.In other words, 1 is the “preserver” in multiplication, It preserves anything multiplies with it .
For example, A*B*0*C = 0 where A, B ,and C are numbers.
![Page 11: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/11.jpg)
Multiplication II
3 copies = 2 copies
* We noted that
![Page 12: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/12.jpg)
Multiplication II
3 copies = 2 copies
so that 3 x 2 = 2 x 3
* We noted that
and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.
![Page 13: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/13.jpg)
Multiplication II
3 copies = 2 copies
so that 3 x 2 = 2 x 3
* We noted that
* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).
and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.
![Page 14: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/14.jpg)
Multiplication II
3 copies = 2 copies
so that 3 x 2 = 2 x 3
* We noted that
* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).
For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.
6 12
and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.
![Page 15: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/15.jpg)
Multiplication II
3 copies = 2 copies
so that 3 x 2 = 2 x 3
* We noted that
* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).
Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish.
For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.
6 12
and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.
![Page 16: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/16.jpg)
Multiplication II
3 copies = 2 copies
so that 3 x 2 = 2 x 3
* We noted that
* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).
Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish.
For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.
6 12
Above observations provide us with short cuts for lengthy multiplication that involves many numbers.
and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.
![Page 17: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/17.jpg)
Multiplication II
3 copies = 2 copies
so that 3 x 2 = 2 x 3
* We noted that
* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).
Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish.
For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.
6 12
Above observations provide us with short cuts for lengthy multiplication that involves many numbers. They also provide ways to double check our answers as shown below.
and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.
![Page 18: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/18.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
![Page 19: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/19.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example, 2 x 4 x 3 x 5 = (2 x 4) x (3 x 5)
![Page 20: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/20.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example, 2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
![Page 21: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/21.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
![Page 22: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/22.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
= 10 x 12 = 120
![Page 23: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/23.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
= 10 x 12 = 120
Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.
![Page 24: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/24.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
= 10 x 12 = 120
Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.
Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25
![Page 25: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/25.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
= 10 x 12 = 120
ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.
Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.
Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25
2 x 4 x 3 x 5 x 25
I.
![Page 26: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/26.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
= 10 x 12 = 120
ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.
Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.
Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25
2 x 4 x 3 x 5 x 25
10 100I.
![Page 27: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/27.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
= 10 x 12 = 120
ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.
Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.
= 3 x 10 x 100= 3,000
Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25
2 x 4 x 3 x 5 x 25
10 100I.
![Page 28: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/28.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
= 10 x 12 = 120
ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.
Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.
= 3 x 10 x 100= 3,000
Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25
2 x 4 x 3 x 5 x 25
10 100
2 x 4 x 3 x 5 x 25
50
20
I. II.
![Page 29: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/29.jpg)
Multiplication IIi. For a lengthy multiplication, multiply in pairs.
For example,
10
2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120
or 2 x 3 x 4 x 5
12
= 10 x 12 = 120
ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.
Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.
= 3 x 10 x 100= 3,000
Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25
2 x 4 x 3 x 5 x 25
10 100
= 20 x 3 x 50= 3,000
2 x 4 x 3 x 5 x 25
50
20
I. II.
![Page 30: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/30.jpg)
Multiplication IIEven if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 31: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/31.jpg)
Multiplication II
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 32: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/32.jpg)
Multiplication II
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 33: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/33.jpg)
Multiplication II
= 9 x 2 x 7 x 2
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 34: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/34.jpg)
Multiplication II
= 9 x 2 x 7 x 2
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
= 18 x 7 x 2
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 35: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/35.jpg)
Multiplication II
= 9 x 2 x 7 x 2
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
= 18 x 7 x 2
= 136 x 2
= 272
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 36: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/36.jpg)
Multiplication II
= 9 x 2 x 7 x 2
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
= 18 x 7 x 2
= 136 x 2
= 272
Doing it in pairs:(3 x 3) x (2 x 7) x 2
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 37: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/37.jpg)
Multiplication II
= 9 x 2 x 7 x 2
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
= 18 x 7 x 2
= 136 x 2
= 272
Doing it in pairs:(3 x 3) x (2 x 7) x 2
= 9 x 14 x 2
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 38: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/38.jpg)
Multiplication II
= 9 x 2 x 7 x 2
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
= 18 x 7 x 2
= 136 x 2
= 272
Doing it in pairs:(3 x 3) x (2 x 7) x 2
= 9 x 14 x 2
= 272
= 9 x 28
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
![Page 39: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/39.jpg)
Multiplication II
= 9 x 2 x 7 x 2
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
= 18 x 7 x 2
= 136 x 2
= 272
Doing it in pairs:(3 x 3) x (2 x 7) x 2
= 9 x 14 x 2
= 272
= 9 x 28
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
2 + 2 + 2 = 3 x 2
3 copies
We simplify the notation forrepetitive additions as:
![Page 40: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/40.jpg)
Multiplication II
We simplify the notation forrepetitive multiplication as:
= 9 x 2 x 7 x 2
b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.
3 x 3 x 2 x 7 x 2Doing it in the order that’s given:
= 18 x 7 x 2
= 136 x 2
= 272
Doing it in pairs:(3 x 3) x (2 x 7) x 2
= 9 x 14 x 2
= 272
= 9 x 28
Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.
2 + 2 + 2 = 3 x 2
3 copies
We simplify the notation forrepetitive additions as:
2 * 2 * 2 = 23 = 8
3 copies
![Page 41: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/41.jpg)
Multiplication IIAbout the Notation
![Page 42: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/42.jpg)
Multiplication II
In the notation
= 2 * 2 * 223 = 8
About the Notation
![Page 43: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/43.jpg)
Multiplication II
In the notation
= 2 * 2 * 223this is the base = 8
About the Notation
![Page 44: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/44.jpg)
Multiplication II
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
= 8
About the Notation
![Page 45: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/45.jpg)
Multiplication II
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”
= 8
About the Notation
![Page 46: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/46.jpg)
Multiplication II
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”
= 8
About the Notation
Recall that for repetitive addition, we write
3 copies
2 + 2 + 2 as 3 x 2 = 3(2) = 2(3) with 3 to one the side.
![Page 47: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/47.jpg)
Multiplication II
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”
= 8
About the Notation
Recall that for repetitive addition, we write
3 copies
So for repetitive multiplication, to distinguish it from addition, we store the number of repetition in the upper corner.
2 + 2 + 2 as 3 x 2 = 3(2) = 2(3) with 3 to one the side.
![Page 48: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/48.jpg)
Multiplication II
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”
= 8
About the Notation
Recall that for repetitive addition, we write
3 copies
3 copies
So for repetitive multiplication, to distinguish it from addition, we store the number of repetition in the upper corner.
2 + 2 + 2 as 3 x 2 = 3(2) = 2(3) with 3 to one the side.
Hence, we write 2 * 2 * 2 as 23.
![Page 49: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/49.jpg)
Multiplication IIExample B. Calculate the following.
a. 3(4) b. 34 c. 43
d. 22 x 3 e. 2 x 32 f. 22 x 33
![Page 50: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/50.jpg)
Multiplication IIExample B. Calculate the following.
= 12
a. 3(4) b. 34 c. 43
d. 22 x 3 e. 2 x 32 f. 22 x 33
![Page 51: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/51.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3a. 3(4) b. 34 c. 43
d. 22 x 3 e. 2 x 32 f. 22 x 33
![Page 52: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/52.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
a. 3(4) b. 34 c. 43
d. 22 x 3 e. 2 x 32 f. 22 x 33
![Page 53: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/53.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
a. 3(4) b. 34 c. 43
d. 22 x 3 e. 2 x 32 f. 22 x 33
![Page 54: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/54.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
d. 22 x 3 e. 2 x 32 f. 22 x 33
![Page 55: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/55.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
d. 22 x 3 e. 2 x 32 f. 22 x 33
![Page 56: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/56.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
= 2*2*3
d. 22 x 3 e. 2 x 32 f. 22 x 33
= 12
![Page 57: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/57.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
= 2*2*3 = 2*3*3
d. 22 x 3 e. 2 x 32 f. 22 x 33
= 12
![Page 58: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/58.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
= 2*2*3 = 2*3*3
= 6*3
= 18
d. 22 x 3 e. 2 x 32 f. 22 x 33
= 12
![Page 59: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/59.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
= 2*2*3 = 2*3*3
= 6*3
= 18
= 2*2*3*3*3
d. 22 x 3 e. 2 x 32 f. 22 x 33
= 12
![Page 60: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/60.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
= 2*2*3 = 2*3*3
= 6*3
= 18
= 2*2*3*3*3
d. 22 x 3 e. 2 x 32 f. 22 x 33
= 4 * 9= 12 * 3
![Page 61: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/61.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
= 2*2*3 = 2*3*3
= 6*3
= 18
= 2*2*3*3*3
d. 22 x 3 e. 2 x 32 f. 22 x 33
= 36 * 3
= 4 * 9= 12 * 3
= 108
![Page 62: 1.7 multiplication ii w](https://reader034.vdocuments.mx/reader034/viewer/2022042815/55629395d8b42a68128b4f1b/html5/thumbnails/62.jpg)
Multiplication IIExample B. Calculate the following.
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
= 2*2*3 = 2*3*3
= 6*3
= 18
= 2*2*3*3*3
d. 22 x 3 e. 2 x 32 f. 22 x 33
= 36 * 3
= 4 * 9= 12 * 3
= 108
Problems d, c and e are the same as 22(3), 2(32), and 22(33).