FoundationsConcept reference

Dividing powers

The quotient of two integer powers of the same nonzero real base is that base raised to the numerator exponent minus the denominator exponent.

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On this page 7 sections

Overview
In 2^4/2^2, two denominator factors cancel two numerator factors
Equal exponents give zero: 4^3/4^3=4^0=1
Check your understanding
Key takeaway
Sources & further reading
Concept connections

Figure 12⁴ divided by 2². Two numerator factors of 2 cancel the two denominator factors; two numerator factors remain.

01In 2^4/2^2, two denominator factors cancel two numerator factors#

In $2^4/2^2$ , two denominator factors cancel two numerator factors. Two factors of 2 remain.

\frac{2^4}{2^2}=2^{4-2}=2^2=4

For the same nonzero base, subtract the denominator’s exponent from the numerator’s exponent:

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

If more factors appear below, the exponent is negative. For example, $3^2/3^5=3^{-3}=1/27$ . Keep the subtraction in numerator-minus-denominator order.

Check your reasoning

Write $7^5/7^2=7^k$ . What is $k$ ?

Show answer and explanation

Subtract the denominator exponent: $5-2=3$ .

02Equal exponents give zero: 4^3/4^3=4^0=1#

Equal exponents give zero: $4^3/4^3=4^0=1$ . The common base must be nonzero so the denominator is not zero.

Check your reasoning

Someone writes $5^2/5^6=5^3$ by dividing exponents. Which exponent repairs it?

A $4$
B $3$
C $-4$

Show answer and explanation

$-4$

Subtract in order: $2-6=-4$ .

03Check your understanding#

Check your reasoning

A new image has width $2^6$ units; its reference width is $2^2$ units. The new-to-reference ratio is $2^k$ . What is $k$ ?

Show answer and explanation

Divide new width by reference width, then subtract $6-2=4$ .

Key takeaway

Keep the common base; subtract the denominator’s exponent.

Rewrite a quotient of powers with the same nonzero base as one power.

Sources & further reading

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

01In 2^4/2^2, two denominator factors cancel two numerator factors#

02Equal exponents give zero: 4^3/4^3=4^0=1#

03Check your understanding#

Keep the common base; subtract the denominator’s exponent.

Sources & further reading

Reference this concept