topic badge
CanadaON
Grade 11

Power of a power with integer bases and negative powers

Lesson

We've already looked at the power of a power rule, which states:

$\left(a^m\right)^n=a^{mn}$(am)n=amn

Now we are going to use apply this rule to terms involving negative bases, and also negative exponents.

Remember!

The base number stays the same. We just multiply the powers.

 

Negative numbers with negative powers 

When we raise a negative numbers by an even power, we get a positive answer e.g. $\left(-4\right)^2=-4\times\left(-4\right)$(4)2=4×(4)$=$=$16$16

The same pattern applies to negative powers: $\left(-2\right)^{-4}=2^{-4}$(2)4=24$=$=$\frac{1}{16}$116

This means we can simplify expressions involving negative numbers with negative exponents, and express them as a fraction with a positive exponent. If the power is even the fraction will be positive, but if the power is odd the fraction will be negative.

For example, $\left(-5\right)^{-3}=\frac{1}{\left(-5\right)^3}$(5)3=1(5)3 $=$= $-\frac{1}{125}$1125

 

Examples

Question 1

Simplify the following into the form $a^b$ab:

$\left(6^7\right)^{-3}$(67)3

Question 2

Evaluate $\left(5^{-9}\right)^0$(59)0.

Question 3

$\left(\left(-4\right)^{-8}\right)^2$((4)8)2 simplifies to which of the following:

  1. $-4^{-6}$46

    A

    $4^{-6}$46

    B

    $4^{-16}$416

    C

    $-4^{-16}$416

    D

Question 4

We want to simplify the following expression using exponent laws: $4^{-2}\times64^{-3}$42×643.

  1. To use the exponent laws, we need both terms to be written with the same base.

    Fill in the box to re-write $64^{-3}$643 with a base of $4$4.

    $64^{-3}=\left(4^{\editable{}}\right)^{-3}$643=(4)3

  2. Using the result of the previous part, express $4^{-2}\times64^{-3}$42×643 in simplest positive exponential form.

 

 

 

Outcomes

11C.A.2.3

Determine, through investigation, the exponent rules for multiplying and dividing numerical expressions involving exponents [e.g., (1/2)^3 x(1/2)^2 ], and the exponent rule for simplifying numerical expressions involving a power of a power [e.g., (5^3)^2 ]

What is Mathspace

About Mathspace