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Exponent Rules

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The explanations and examples below on exponent rules follow on from the Power (Exponents and Bases) page which you might want to start with.

The examples on simplifying exponential expressions show when and how the above exponent rules are used.

Exponents: The Product Rule

The product rule for exponents states that when we multiply exponential expressions having the same base, we can add the exponents and keep the base unchanged.

Product Rule

If m and n are natural numbers, and a is a real number, then

am x an = am + n

Example

Rewrite 4243 using a single base and exponent.

The product rule states that am x an = am + n

Applying the rule to the expression and simplifying we get

42 x 43 = 42 + 3 = 45

Think about it this way

42 = 4 x 4 and 43 = 4 x 4 x 4

so

42 x 43 = (4 x 4) x (4 x 4 x 4) = 45

Zero as an exponent

According to the zero exponent rule, any nonzero number raised to the power 0 equals 1.

Zero Rule

If a is any nonzero number, then

a0 = 1

Example

Evaluate the expression (-2)0

The zero exponent rule states that a0 = 1

Applying the rule to the expression we get

(-2)0 = 1

Quotient rule for exponents

We make use of quotient rule when dividing exponents with the same base. The quotient rule for exponents states that if we divide exponents with the same base, then we can subtract the exponents and keep the base unchanged.

Quotient Rule

If b is any nonzero real number, and m and n are nonzero integers, then

bm = bm - n
bn
Example
Simplify the expression 44 - 24
42 23

The exponents have the same base so we can directly apply the quotient rule and rewrite the given expression as shown below.

44 - 24

= 44-2 - 24-3 = 42 - 21

42 23

 

42 - 21 = 16 - 2 = 14

 

44 - 24

simplifies to 14

42 23

Negative Exponent Rule

We use the negative exponent rule to change an expression with a negative exponent to an equivalent expression with a positive exponent. The rule states that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power. In other words, an expression raised to a negative exponent is equal to 1 divided by the expression with the sign of the exponent changed.

Negative Exponent Rule

For any nonzero real number b, and any whole number m,

b-m = 1
bm

 

Example

Write the expression 6a-2 without a negative exponent.

Apply the rule to rewrite the term a-2 since a is raised to a negative power. Write the reciprocal of a and raise it to the opposite power of –2, which is 2.

6a–2 = 6 x
1 = 6
a2 a2
So, 6a-2 written without negative exponent is 6
a2

 

Simplifying Exponential Expressions - Putting it altogether

We can simplify exponential expressions using a suitable combination of the rules and properties above.

Question: Simplify (5x-2y2)-3
(x4y-3)-4

Step 1

We know that the power rule for exponents states that if x is a real number, and m and n are integers,
then (xm)n = xmn

1. Apply the power rule to both the numerator and the denominator, and then simplify.

(5x-2y2)-3 = 5-3x-2.-3y2.-3
(x4y-3)-4 x4.-4y-3.-4
  = 5-3x6y-6
  x-16y12

Notation: a dot is used above to represent multiplication

Step 2

The quotient rule for exponents states that if m and n are natural numbers, and a is a real number, then

am = a m - n
an

 

2. Rewrite the expression using the quotient rule for exponents.

5-3x6y-6 = 5-3x6-(-16)y-6-12
x-16y12
 

= 5-3x6+16y-6-12

  = 5-3x22y-18

Step 3

The negative exponent rule states that for any nonzero real number, a, and any whole number m, then

a-m = 1
am

 

3. Use the negative exponent rule to rewrite the terms without negative exponents and simplify.

= 5-3x22y-18 = x22
  53y18
  = x22
  125y18
(5x-2y2)-3 simplifies to x22
(x4y-3)-4 125y18
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