Chapter P

Prerequisites: Fundamental Concepts of Algebra

P.2 Exponents and Scientific Notation

ObjectivesAt the end of this session, you will be able to:

Understand and use integer exponents.Use properties of exponents.Simplify exponential expressions.Use scientific notation.

Index1. Exponents2. Properties of Exponents

2.1 Zero Exponent Rule2.2 Negative Exponent Rule2.3 Product Rule2.4 Quotient Rule2.5 Power Rule2.6 Products to Power Rule2.7 Quotients to Powers Rule

3. Simplifying Exponential Expressions4. Word of Caution!5. Scientific Notation6. Applications7. Summary

Exponents are used everywhere in algebra and beyond. This section covers the basic definition and rules of exponents.  Exponents are another way to write multiplication.

For example: 3 . 3 . 3 . 3 can be written as 34 and it is read as “three raised to the power four” or as “fourth power of three”.

In 34, the integer 3 is called the base and 4 is called the exponent. Writing the product of a number by itself several times in this form is known as

exponential notation or power notation.Exponential notation is useful in situations where the same number is multiplied repeatedly. The number being multiplied is called the base, and the exponent tells how many times the base is multiplied by itself.

The base can be any sort of number - a whole number, a decimal number, or a fraction can all be raised to a power.

A number with an exponent is said to be "raised to the power" of that exponent. Thus, we can state the following formal definition of exponents:Exponent:If x is a real number and n is a natural number, then,

xn = x . x . x . x . … .x

1. Exponents

Base x appears as a factor n times.

Exponent

NOTE: If the base is a negative number, then parentheses should be used to indicate this.Example: (-3)(-3)(-3)(-3)(-3)(-3) = (-3)6

If parentheses is not used, the number is expressed as –36 and this means something quite different. It is the negative of the number 36

(-3)6 = (-3)(-3)(-3)(-3)(-3)(-3) = 729, but-36 = - (3)(3)(3)(3)(3)(3) = -729 so be careful with negative

signs !Also -36 is read as “ the opposite of 3 to the sixth power.” By contrast, (-3)6 is read as “ negative 3 to the sixth power.”Special Cases:

Any number raised to the power of one equals itselfExample: 231 = 23.

A number with an exponent of two is referred to as the square of a number. Example: 32 The square of a whole number is known as a perfect square. The numbers 1, 4, 9, 16, and 25 are all perfect squares.

Example: 12=1, 22=4, 32=9, 42=16 and so on. A number with an exponent of three is referred to as the cube of a number.

Example: 53

The cube of a whole number is known as a perfect cube. The numbers 1, 8, 27, 64, and 125 are all perfect cubes.

Example: 13=1, 23=8, 33=27,43=64 and so on.

1. Exponents (Cont…)

1. Exponents (Cont…)Evaluating an Exponential Expression:

Example1: (-4)3 . 22

(-4)3 . 22 = (-4). (-4). (-4). 2 . 2 = - 64 . 4 = - 256

Example 2:

3

3

1

5

1 1 1 1 1. . (We write the base - in a product 3 times)

5 5 5 5 5

1 =

125

2.1 ZERO AS AN EXPONENT:Zero can also be used as an exponent. Any number raised to the zero power (except 0) equals 1.

That is, except for 0, any base raised to the 0 power simplifies to be the number 1. Example:

Note that the exponent doesn’t become 1, but the whole expression simplifies to be the number 1.

Now let us evaluate certain expressions with zero as exponents: Evaluate:

i. (7xyz)0

(7xyz)0 = 1 (Any expression raised to the power zero simplifies to be 1)ii. -15x0

Be careful with this example ! According to the order of operations, evaluate exponents before doing any multiplication. This means we need to find x raised to the 0 power first and then multiply it by -15. -15x0 = -15 . (1) (x0 = 1) = -15

Thus, we can state the following rule:

Zero Exponent Rule:If x is any real number other than 0, then x0 = 1

2. Properties of Exponents

Base0 Value

20 = 1

(-6)0 = 1

-80 = -(80 ) = -(1) = -1

2. Properties of Exponents (Cont…)2.2 Negative Exponents:

Negative numbers as exponents have a special meaning. The rule is as follows:

The Negative Exponent Rule:If x is any real number other than 0 and n is natural number, then

Since exponents are another way to write multiplication, in the case of a negative exponent, to write it as a positive exponent we do the multiplicative inverse. This means we take the reciprocal of the base.

Evaluating expressions containing negative exponents: Evaluate:

i. 4-1  1 1

4 (Re write with a positive exponent by taking reciprocal of the base)4

n nn n

1 1x or x

x x

2. Properties of Exponents (Cont…)ii.

iii.

iv.

2

2 n2 n

2

5

1 15 (Negative Exponent Rule : x )

( 5) x

1(Usin g the definition of exponents ( 5) 25)

25

2

2 n2 n

2

8

1 18 (Negative Exponent Rule : x )

8 x

1 (Usin g the definition of exponents 8 8.8 64)

64

5

5 5n

n

n5 n

5 n

5 5

1

2

1 2 1(Negtaive Exponent Rule : x )

2 1 x

2 x x (Quotient Rule : )

1 y y

=32 (Using the definition of exponents: 2 =32 and 1 =1)

2.3 The Product Rule:Let us first start by using the definition of exponents to help understand how we get to the law for multiplying like bases with exponents. Consider the expression x2 . X3 It can be written as follows:

x2 . x3 = (x . x) . (x . x . x)Now the expression (x . x) . (x . x . x) can be simplified to x . x. x . x . x, which is x5

Note that 2 + 3 = 5, which is the exponent we ended up with. The expression was a product of 2 xs and 3 xs. And a total of 5 xs in the product. 

This multiplication rule tells us that we can simply add the exponents when multiplying two exponential expressions with the same base.

Let us put this idea together into a general rule:

The Product Rule:xm . xn = xm+n

That is, when we multiply exponential expressions with the same base, we add the exponents.

Evaluating exponential expressions using product rule: Use the product rule to simplify each expression:

i. a6 . a10

a6 . a10 = a6+10 (Using the product rule, xm . xn = xm+n) = a16

ii. 24 . 2-7

24 . 2-7 = 24+(-7) (Using the product rule, xm . xn = xm+n) = 24 – 7 = 2-3

2. Properties of Exponents (Cont…)

3

1 1=

2 8

2.4 The Quotient Rule:Let us start by using the definition of exponents to help understand how we get to the rule of dividing like bases with exponents. Consider the expression below:

Note that 5 - 2 = 3, the final answer’s exponent. When we divide like bases we subtract the exponent in the denominator from the exponent in the numerator.Thus, we can put this idea together into a general rule:

The Quotient Rule:

When we divide exponential expressions with the same non-zero base, we subtract the exponent in the denominator from the exponent in the numerator. Then, we use this difference as the exponent of the common base.

NOTE: Always keep in mind that we always take the numerator’s exponent minus the denominator’s exponent.  

53

2

x x.x.x.x.x = =x.x.x=x

x x.x

mm-n

n

x =x ,x 0

x

2. Properties of Exponents (Cont…)

2. Properties of Exponents (Cont…)Evaluating exponential expressions using the quotient rule:

Use the quotient rule to simplify each expression:i.

ii.

NOTE: represent different numbers.

6

10

66 - 10

10

-44

a

a

a=a (Using the quotient rule)

a1

=a = a

2 5

5 2

2 2 and

2 22

2-5 35 3

55 2 3

4

2 1 1 2 2

2 2 8

22 2 8

2

2

3

22 ( 3)

3

2 + 3

5

2

22

=2 (Using the quotient rule)2 =2 (minus.minus = plus)

=2 = 32

2.5 The Power Rule:

This property is applied when an expression containing a power is itself raised to a power.

Now let us understand how we get to the rule for raising a base to two exponents:

Consider (x2)3 = (x2) . (x2) . (x2) (Using the definition of exponents) = x2+2+2 (Using product rule: (xm). (xn) = xm+n) = x6

Note that 2 times 3 is 6, that is, 2 . 3 = 6, which is the exponent of the final answer. We can think of this as 3 groups of 2, which would come out to be 6.

In other words, when we raise a base to two exponents, we multiply those exponents together.

Hence we have the following rule:

The Power Rule:

(xm)n = xm.n

When an exponential expression is raised to a power, we multiply the exponents.

Evaluating exponential expressions using the power rule: Use the power rule to simplify each expression:

2. Properties of Exponents (Cont…)

Example : (x7)5

= (x7)5 = x(7) . (5)

= x35

Example : 3 1 (3).( 1)

-3

3

(4 ) = 4

= 4

1 1 = =

4 64

2.6 Products to Power Rule:This property of exponents is applied when we are raising a product to a power Consider (xy)3 = (xy) . (xy) . (xy) (Using the definition of exponents )

= (x . x . x) . (y . y. y) (Grouping like terms together) = x3 . y3

In other words we can say, when we have a product (not a sum or difference) raised to an exponent, we can simplify by raising each base in the product to that exponent. Thus, we have the following rule:

Products to Power:(xy)n = xn . yn

When a product is raised to a power, raise each factor of the product to that power.

Simplify:i. (xy)8

(xy)8 = x8 . y8 (Using products to power rule: (xy)n = xn . yn)

ii. (-4xyz)3

(-4xyz)3 = (-4)3 . x3 . y3 . z3 (Using products to power rule: (xy)n = xn . yn) = - 64 x3y3z3

2. Properties of Exponents (Cont…)

2.7 Quotients to Powers Rule:

This exponential property is applied when we are raising a quotient to power. Consider

Since division is really multiplication of the reciprocal, it has the same basic idea as when we raise a product to an exponent.

In other words, when we have a quotient (not a sum or difference) raised to an exponent, we can simplify by raising each base in the numerator and denominator of the quotient to that exponent.

Thus, we have the following rule:

Quotients to Powers Rule:

When a quotient is raised to a power, raise the numerator to that power and divide by the

denominator raised to that power.

n n

n

x x, y 0

y y

2. Properties of Exponents (Cont…)

3 3

3

x x x x x.x.x x. .

y y y y y.y.y y

2. Properties of Exponents (Cont…)Now let us simplify some exponential expressions by raising each quotient to the given power:

i.

ii.

Up to now, we have applied a single property of exponents to simplify given exponential expressions. Next, we will be using more than one property to simplify an exponential expression.

2

2 n2 n

2 n

2

4

y

4 ( 4) x x (Using the quotients to powers rule: = ,y 0)

y y y y

16 =

y

3

n3 3 n

3 n

3

4

3 3 x x (Using the quotients to powers rule: = ,y 0)

4 4 y y

27 =

64

3. Simplifying Exponential Expressions

When simplifying an exponential expression, we write it so that each base is written one time with one positive exponent. In other words, we write it in the most condensed form making sure that all the exponents are positive.

Many times we have to use more than one rule to simplify exponential expressions. As long as we use the rule appropriately, its fine. 

First let us summarize the properties of exponents we have covered so far:1. x0 = 12.

3. xm . xn = xm + n

4.

5. (xm)n = xm . n

6. (xy)n = xn . yn

7.

n nn n

1 1x or x

x x

n n

n

x x, y 0

y y

mm n

n

x =x ,x 0

x

3. Simplifying Exponential Expressions (Cont…)

Example 1: Simplify (6x5y0z-3)-2

(6x5y0z-3)-2 = 6-2 . x(5)(–2). y(0)(–2) . z(-3)(–2) (Using (xy)n = xn . yn) = 6-2 . x–10. y0 . z6 (Using (xm)n = xm.n)

Example 2: Simplify

6 -n 02 10 n

610

6

10

1 1 1. .(1).z (Using x = and x = 1)

6 x x1 1

. .(1).z36 x

z

36x

12 2

16 4

12 2 12 2

16 4 16 4

m12-16 2-(-4)

100x y

20x y

100x y 100 x y. . (Grouping factors with the same base)

20x y 20 x y

x = 5. (x ).(y ) (Using

x

m-nn

-4 2+4

6

4

=x ,x 0)

= 5.(x ).(y ) (minus. minus = plus)

5.y =

x

4. Avoid This!Try to avoid the following errors that can occur when simplifying exponential expressions:Correct Incorrect Explanation

26 = 2 . 2 . 2 . 2 . 2 . 2 =64 26 = 2 . 6 = 12 Do not multiply the base and the exponent. 26 is not equal to 12, it's 64!

x3 . x4 = x3+4 = x7 x3 . x4 = x12 The exponents should be added, not multiplied.

23 . 25 = 28 23 . 25 = 48 The common base should be retained, not multiplied.

42 . 23 = (4) . (4) . (2) . (2) . (2) =128 42 . 23 = 82+3 = 85 = 32,768 The multiplication rule only applies to expressions with the same base. Four squared times two cubed is not the same as 8 raised to the power two plus three.

4. Avoid This!

Correct Incorrect Explanation

22 + 23 = (2) . (2) + (2) . (2) . (2) = 4 + 8 = 12

22 + 23 = 22+3 = 25 = 32

The multiplication rule applies just to the product, not to the sum of two numbers.

The exponents should be subtracted, not divided.

(2x)3 = 8x3 (2x)3 = 2x3 Both factors should be cubed.

Only the exponent should change sign.

The exponent applies to the entire expression x + y.

1616 4 12

4

55 5

5

1616/ 4 4

4

55 5

5

nn

1b

b n

n

1b

b

1 1(x y)

(x y)

1 1 1

(x y)x y

5. Scientific NotationThis section takes a look at the basic definition of scientific notation, an application that involves writing the number using an exponent on base 10. Since part of a number that is written in scientific notation is 10 raised to a power, when we multiply or divide these types of numbers we need to remember some of our exponent rules.Scientific Notation is a way to express very small or very large numbers. It is most often used in "scientific" calculations where the analysis must be very precise.Scientific Notation consists of two parts:

(1) a number between 1 and 10               

(2) a power of 10.NOTE: A large or small number may be written as any power of 10. However, correct scientific notation must satisfy the above criteria.

3.2 x 1013 is correct scientific notation.23.6 x 10-8 is incorrect scientific notation.

Remember that the first digit to the left of decimal point must be greater than or equal to 1 and less than 10.IMPORTANT: It is customary to use the multiplication symbol, ‘x’ , rather than a ‘.’, to indicate multiplication in scientific notation.Thus, we can state the following generalization:Scientific Notation:A number is written in scientific notation, if it is written in the form:

a x 10n where and n is an integer to the power of 10.

1 n < 10

1 n < 10

5. Scientific Notation (Cont…)Example 1: The speed of light is 300000000 meters/sec.

We may write, 300000000 = 3 x 108

Thus, in scientific notation, we express the speed of light as 3 x 108 m/sec.

Example 2: The velocity of sound is 33000 cm/sec.

We write, 33000 = 3.3 x10000 = 3.3 x 104

Thus, in scientific notation, we express the velocity of sound as 3.3 x 104 cm/sec.

Writing a Number in Scientific Notation: STEP 1: Place decimal point such that there is one non-zero digit to the left of

the decimal point.

In other words, we will put the decimal after the first non zero number.  STEP 2: Count number of decimal places the decimal has "moved" from the

original number. This will be the exponent of the 10.

We count the number of decimal places moved in Step 1 .

If the decimal point was moved to the left, the exponent is positive.

If the decimal point is moved to the right, the exponent is negative. STEP 3:Write as a product of the number (found in Step 1) and 10 raised to the

power of the exponent (found in Step 2).

If the original number was less than 1, the exponent is negative; if the original number was greater than 1, the exponent is positive.

 

5. Scientific Notation (Cont…)Example 1: Write the number in scientific notation:   4,750,000.

Step 1: Place decimal point such that there is one non-zero digit to the left of the decimal point.

4,750,000. (Decimal is at the end of the number)

4 . 750000 (We put the decimal after the first non-zero number)

Step 2: Count number of decimal places the decimal has "moved" from the original number. This will be the exponent of the 10.

We started at the end of the number 4,750,000 and moved between 4 and 7, that is, we moved 6 places. We moved the decimal to the left, so the exponent is positive.

So, our exponent is +6.

NOTE: The original number is greater than 1, so the exponent is positive. Step 3: Write as a product of the number (found in Step 1) and 10 raised to the

power of the exponent (found in Step 2).

4.75 x 106

5. Scientific Notation (Cont…)Example 2: Write the number in scientific notation:   .000015.

Step 1: Place decimal point such that there is one non-zero digit to the left of the decimal point.

.000015 (Decimal is at the beginning of the number)

1.5 (We put the decimal after the first non-zero number)

Step 2: Count number of decimal places the decimal has "moved" from the original number. This will be the exponent of the 10.

We started at the beginning of the number .000015 and moved between 1 and 5, that is, we moved 5 places. We moved the decimal to the right, so the exponent is negative.

So, our exponent is -5.

NOTE: The original number is less than 1, so the exponent is negative. Step 3: Write as a product of the number (found in Step 1) and 10 raised to the

power of the count (found in Step 2).

1.5 x 10-5

5. Scientific Notation (Cont…)Write a Scientific Number in Standard Form:For converting a number in scientific notation to standard form, we basically just multiply the first number times the power of 10.  Whenever we multiply by a power of 10, we are actually moving the decimal place.

Move decimal point to right for positive exponent of 10. Move decimal point to left for negative exponent of 10. Make sure you add in any zeros that are needed.

Example 1: Write the number in decimal notation without exponents: 5.32 x 107

5.32 x 107 = 53200000 (Move the decimal 7 places to the right)Example 2: Write the number in decimal notation without exponents: 3.15 x 10-4

3.15 x 10-4 = 0.000315 (Move the decimal 4 places to the left)

NOTE: A negative on an exponent and a negative on a number mean two different things. Do not confuse these! For example:

-0.00036 = -3.6 x 10-4

0.00036 = 3.6 x 10-4

36,000 = 3.6 x 104

-36,000 = -3.6 x 104

 .

5. Scientific Notation (Cont…)Multiplication and Division of scientific notation:To Multiply and Divide using Scientific Notation:

STEP 1: Multiply/divide decimal numbers with each other.  STEP 2: Use exponent rules to "combine" powers of 10.  STEP 3: If not "correct" scientific notation, change accordingly.

Example: Perform the indicated operation and express the answer in decimal form.(3.2 x 103)(5.1 x 10-5)

(3.2 x 103)(5.1 x 10-5) = (3.2 x 5.1)(103 x 10-5) (Step 1) = 16.32 x 103-5

= 16.32 x 10-2

= .1632 (Move decimal 2 places to the left)Example: Perform the indicated operation and express the answer in decimal form.

30

25

30 30

25 25

30 25

5

9 x 10

3 x 10

9 x 10 9 10 x

3 x 10 3 10

3 x 10

3 x 10

300000

6. ApplicationsThe mass of one oxygen molecule is 5.3 x 10-23 gram. Find the mass of 20,000 molecules of oxygen. Express the answer in scientific notation.

Solution:

Mass of 1 oxygen molecule

= 5.3 x 10-23 gram

Mass of 20,000 molecules of oxygen

= (5.3 x 10-23) . (20,000) grams

= 5.3 x 10-23) . (2 x 10 4) grams (Writing 20,000 in scientific notation)

= (5.3 x 2) x (10-23 x 10 4) grams

= 10.6 x 10-23+4 grams (Using the product rule (xm)n=xm+n)

= 10.6 x 10-19 grams

= 1.06 x 10-18 grams (Writing the answer in the scientific notation)

7. SummaryLet us recall what we have studied so far:

Exponent: If x is a real number and n is a natural number, then,

xn = x . x . x . x . … .x

multiplied n times

where x is the base and n is the exponent.

Properties of exponents:

Property Definition

Zero exponent rule x0 = 1

Negative exponent rule

Product rule xm . xn = xm+n

Quotient rule

Power rule (xm)n = xm.n

Products to power rule (xy)n = xn . yn

Quotients to power rule

n nn n

1 1x or x

x x

n n

n

x x, y 0

y y

mm n

n

x =x ,x 0

x

7. Summary (Cont…)Scientific Notation:A number can be written in scientific notation by changing it to the following form: 

a x 10n where 1 < a < 10 and n is an integer power of 10.

Writing a Number in Scientific Notation: STEP 1: Place decimal point such that there is one non-zero digit to the left of the

decimal point. STEP 2: Count number of decimal places the decimal has "moved" from the

original number. This will be the exponent of the 10. STEP 3: Write as a product of the number (found in Step 1) and 10 raised to the

power of the exponent (found in Step 2).