52 pythagorean theorem and square roots

A right triangle is a triangle with a right angle as one of its angle.

Pythagorean Theorem and Square Roots

A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse,

Pythagorean Theorem and Square Roots

hypotenuseC

A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, the two sides A and B forming the right angle are called the legs.

Pythagorean Theorem and Square Roots

hypotenuse

legs

A

BC

A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, the two sides A and B forming the right angle are called the legs.

Pythagorean TheoremGiven a right triangle as shown and A, B, and C be the length of the sides, then A2 + B2 = C2.

Pythagorean Theorem and Square Roots

hypotenuse

legs

A

BC

Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us to compute a length, i.e. a distance, without measuring it directly.

Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us to compute a length, i.e. a distance, without measuring it directly.

Example A. A 5–meter ladder leans against a wall as shown. Its base is 3 meters from the wall. How high is the wall?

5 m

3 m

?

Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us to compute a length, i.e. a distance, without measuring it directly.

Example A. A 5–meter ladder leans against a wall as shown. Its base is 3 meters from the wall. How high is the wall?

5 m

3 m

? = h

Let h be the height of the wall.

Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us to compute a length, i.e. a distance, without measuring it directly.

Example A. A 5–meter ladder leans against a wall as shown. Its base is 3 meters from the wall. How high is the wall?

5 m

3 m

? = h

Let h be the height of the wall.The wall and the ground form a right triangle, hence by the Pythagorean Theoremwe have that h2 + 32 = 52

Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us to compute a length, i.e. a distance, without measuring it directly.

Example A. A 5–meter ladder leans against a wall as shown. Its base is 3 meters from the wall. How high is the wall?

5 m

3 m

? = h

Let h be the height of the wall.The wall and the ground form a right triangle, hence by the Pythagorean Theoremwe have that h2 + 32 = 52

h2 + 9 = 25

Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us to compute a length, i.e. a distance, without measuring it directly.

Example A. A 5–meter ladder leans against a wall as shown. Its base is 3 meters from the wall. How high is the wall?

5 m

3 m

? = h

Let h be the height of the wall.The wall and the ground form a right triangle, hence by the Pythagorean Theoremwe have that h2 + 32 = 52

h2 + 9 = 25 –9 –9

subtract 9from both sides

Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us to compute a length, i.e. a distance, without measuring it directly.

Example A. A 5–meter ladder leans against a wall as shown. Its base is 3 meters from the wall. How high is the wall?

5 m

3 m

? = h

Let h be the height of the wall.The wall and the ground form a right triangle, hence by the Pythagorean Theoremwe have that h2 + 32 = 52

h2 + 9 = 25 –9 –9h2 = 16

subtract 9from both sides

Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us to compute a length, i.e. a distance, without measuring it directly.

Example A. A 5–meter ladder leans against a wall as shown. Its base is 3 meters from the wall. How high is the wall?

5 m

3 m

? = h

Let h be the height of the wall.The wall and the ground form a right triangle, hence by the Pythagorean Theoremwe have that h2 + 32 = 52

h2 + 9 = 25 –9 –9h2 = 16

By trying different numbers for h, we find that 42 = 16 so h = 4 or that the wall is 4–meter high.

subtract 9from both sides

Pythagorean Theorem and Square RootsSquare Root

Pythagorean Theorem and Square RootsSquare RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”,

Pythagorean Theorem and Square RootsSquare RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square RootsSquare RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square RootsSquare RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Note that both +4 and –4, when squared, give 16.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4. We refer “–4” as the “negative of the square root of 16”.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the square root of x.

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4. We refer “–4” as the “negative of the square root of 16”.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the square root of x. This is written as sqrt(x) = a, or x = a.

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4. We refer “–4” as the “negative of the square root of 16”.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) =c.3 =

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 =d. –3 =

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4. We refer “–4” as the “negative of the square root of 16”.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) = 4c.3 =

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 =d. –3 =

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4. We refer “–4” as the “negative of the square root of 16”.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) = 4c.3 =

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 = 1/3d. –3 =

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4. We refer “–4” as the “negative of the square root of 16”.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) = 4c.3 = 1.732.. by calculatoror that 3 ≈ 1.7 (approx.)

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 = 1/3d. –3 =

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4. We refer “–4” as the “negative of the square root of 16”.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”, i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) = 4c.3 = 1.732.. by calculatoror that 3 ≈ 1.7 (approx.)

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 = 1/3d. –3 = doesn’t exist (why?),and the calculator returns “Error”.

Note that both +4 and –4, when squared, give 16. But we designate the “square root of 16” i.e. 16 or sqrt(16) to be +4. We refer “–4” as the “negative of the square root of 16”.

Square RootFrom example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table.

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36,

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36, so we estimate that30 5.5.

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36, so we estimate that30 5.5.In fact 30 5.47722….

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg.

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg. Example B. Find the missing side of the following right triangles.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg. Example B. Find the missing side of the following right triangles.

we are to find the hypotenuse,

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg. Example B. Find the missing side of the following right triangles.

we are to find the hypotenuse, so 122 + 52 = c2

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg. Example B. Find the missing side of the following right triangles.

we are to find the hypotenuse, so 122 + 52 = c2

144 + 25 = c2

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg. Example B. Find the missing side of the following right triangles.

we are to find the hypotenuse, so 122 + 52 = c2

144 + 25 = c2

169 = c2

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg. Example B. Find the missing side of the following right triangles.

we are to find the hypotenuse, so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg. Example B. Find the missing side of the following right triangles.

b. a = 5, c = 12,

we are to find the hypotenuse, so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg.

so 52 + b2 = 122

Example B. Find the missing side of the following right triangles.

b. a = 5, c = 12, we are to find a leg,

we are to find the hypotenuse, so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg.

so 52 + b2 = 122

25 + b2 = 144

Example B. Find the missing side of the following right triangles.

b. a = 5, c = 12, we are to find a leg,

we are to find the hypotenuse, so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg.

so 52 + b2 = 122

25 + b2 = 144b2 = 144 – 25 = 119

Example B. Find the missing side of the following right triangles.

b. a = 5, c = 12, we are to find a leg,

we are to find the hypotenuse, so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions of calculation based on the Pythagorean Theorem –finding the hypotenuse versus finding a leg.

so 52 + b2 = 122

25 + b2 = 144b2 = 144 – 25 = 119Hence b = 119 10.9.

Example B. Find the missing side of the following right triangles.

b. a = 5, c = 12, we are to find a leg,

we are to find the hypotenuse, so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

The number 2 is the length of the hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

The number 2 is the length of the hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers.

The number 2 is the length of the hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers.

The number 2 is the length of the hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers. Most real numbers are irrational, not fractions, i.e. they can’t be represented as ratios of two integers.

The number 2 is the length of the hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers. Most real numbers are irrational, not fractions, i.e. they can’t be represented as ratios of two integers. The real line is populated sparsely by fractional locations.

The number 2 is the length of the hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers. Most real numbers are irrational, not fractions, i.e. they can’t be represented as ratios of two integers. The real line is populated sparsely by fractional locations. The Pythagorean school of the ancient Greeks had believed that all the measurable quantities in the universe are fractional quantities. The “discovery” of these extra irrational numbers caused a profound intellectual crisis.

The number 2 is the length of the hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers. Most real numbers are irrational, not fractions, i.e. they can’t be represented as ratios of two integers. The real line is populated sparsely by fractional locations. The Pythagorean school of the ancient Greeks had believed that all the measurable quantities in the universe are fractional quantities. The “discovery” of these extra irrational numbers caused a profound intellectual crisis. It wasn’t until the last two centuries that mathematicians clarified the strange questions “How many and what kind of numbers are there?”

Pythagorean Theorem and Square Roots

x

3

4

Exercise C. Solve for x. Give the square–root answer and approximate answers to the tenth place using a calculator.

1.4

3

x2. x

12

53.

x

1

14. 2

1

x5. 6

x

6.10

1. sqrt(0) = 2. 1 =

Exercise A. find the following square–root (no calculator).

3. 25 3. 100

5. sqrt(1/9) = 6. sqrt(1/16) = 7. sqrt(4/49)

Exercise A. Give the approximate answers to the tenth place using a calculator.

1. sqrt(2) = 2. 3 = 3. 10 3. 0.6