FoundationsConcept reference

Multiplying powers

The product of two integer powers of the same nonzero real base is that base raised to the sum of their exponents.

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On this page 7 sections

Overview
The product 2^2·2^3 joins two factors of 2 to three more
Adding exponents requires the same base
Check your understanding
Key takeaway
Sources & further reading
Concept connections

01The product 2^2·2^3 joins two factors of 2 to three more#

The product $2^2\cdot2^3$ joins two factors of 2 to three more. Count the five factors.

Figure 1Two factors of 2 followed by three factors of 2 make five factors: 2² × 2³ = 2⁵.

The common base stays the same. Add the exponents to count all the factors:

a^m a^n=a^{m+n}

This rule holds for integer exponents when the common base is nonzero. For example, $5^{-2}\cdot5^3=5^{1}=5$ . Two reciprocal factors cancel two of the three factors.

Check your reasoning

Write $4^3\cdot4^2=4^k$ . What is $k$ ?

Show answer and explanation

There are $3+2=5$ factors of the same base.

02Adding exponents requires the same base#

Adding exponents requires the same base. That rule cannot combine $2^2\cdot3^3$ because the bases 2 and 3 differ.

Check your reasoning

Repair the exponent: $6^2\cdot6^3=6^6$ .

A $6$
B $5$
C $1$

Show answer and explanation

$5$

Add exponents: $2+3=5$ .

03Check your understanding#

Check your reasoning

Two scaling steps multiply a value by $2^4$ and then $2^3$ . Their combined factor is $2^k$ . What is $k$ ?

Show answer and explanation

The factors multiply, so add their exponents: $4+3=7$ .

Key takeaway

Keep the common base; add the exponents when multiplying.

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

Sources & further reading

[1]
OpenStax Elementary Algebra 2e: Use multiplication properties of exponents ↗OpenStax · Article

01The product 2^2·2^3 joins two factors of 2 to three more#

02Adding exponents requires the same base#

03Check your understanding#

Keep the common base; add the exponents when multiplying.

Sources & further reading

Reference this concept