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Negative exponents

A nonzero real number raised to a negative integer power is the reciprocal of its corresponding positive power.

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On this page 7 sections

Overview
Keep dividing by 2 as the exponent decreases
Keep the sign of the base inside parentheses
Check your understanding
Key takeaway
Sources & further reading
Concept connections

01Keep dividing by 2 as the exponent decreases#

Keep dividing by 2 as the exponent decreases. After $2^0=1$ , the next values are one half and one quarter.

Figure 1On y=2^x, exponents −2,−1,0 give 1/4,1/2,1. Moving one exponent step left divides the output by 2.

A negative exponent asks for a reciprocal. For a nonzero base $a$ and positive integer $n$ :

a^{-n}=\frac{1}{a^n}

For $4^{-2}$ , first compute $4^2=16$ . Then take its reciprocal: $4^{-2}=1/16$ . The value is positive; the minus belongs to the exponent.

Check your reasoning

Evaluate $2^{-5}$ . Enter a fraction or decimal.

Show answer and explanation

0.03125

Compute $2^5=32$ , then take the reciprocal: $1/32$ .

02Keep the sign of the base inside parentheses#

Keep the sign of the base inside parentheses. Since $(-2)^3=-8$ , its reciprocal is $(-2)^{-3}=-1/8$ . A zero base is excluded because its reciprocal would divide by zero.

Check your reasoning

Someone changes $5^{-2}$ to $-25$ . Which correction works?

A $1/25$
B $-1/25$
C $25$

Show answer and explanation

$1/25$

A negative exponent takes the reciprocal: $1/5^2=1/25$ .

03Check your understanding#

Check your reasoning

An image uses the scale factor $2^{-3}$ . What fraction of the original width is the new width?

Show answer and explanation

0.125

The multiplier is $1/2^3=1/8$ .

Key takeaway

A negative exponent takes the reciprocal of the whole power.

Evaluate a nonzero base raised to a negative integer power.

Sources & further reading

[1]
OpenStax Elementary Algebra 2e: Integer exponents and scientific notation ↗OpenStax · Article

01Keep dividing by 2 as the exponent decreases#

02Keep the sign of the base inside parentheses#

03Check your understanding#

A negative exponent takes the reciprocal of the whole power.

Sources & further reading

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