Question

explain how to use patterns to show that the expression ((b^5)^3) is equivalent to (b^{15}). verify your answer using properties of exponents.
by the definition of an exponent, ((b^5)^3) can be written as the product of (\boldsymbol{\blacktriangledown}) factors of (b^5). the latter of these can be written as the product of (\boldsymbol{\blacktriangledown}) factors of (b). in all, there are thus (\boldsymbol{\blacktriangledown}) factors of (b), which can be written as (b^{15}). this result can be verified by using the (\boldsymbol{\blacktriangledown}) property.

Explanation:

Step1: Expand using exponent definition

$(b^5)^3 = b^5 \times b^5 \times b^5$

Step2: Expand each $b^5$ term

$b^5 \times b^5 \times b^5 = (b \times b \times b \times b \times b) \times (b \times b \times b \times b \times b) \times (b \times b \times b \times b \times b)$

Step3: Count total $b$ factors

There are $5+5+5=15$ factors of $b$, so this equals $b^{15}$

Step4: Name the exponent property

This uses the Power of a Power Property: $(x^m)^n = x^{m \times n}$

Answer:

By the definition of an exponent, $(b^5)^3$ can be written as the product of $\boldsymbol{3}$ factors of $\boldsymbol{b^5}$. The latter of these can be written as the product of $\boldsymbol{5}$ factors of $\boldsymbol{b}$ each. In all, there are thus $\boldsymbol{15}$ factors of $\boldsymbol{b}$ which can be written as $b^{15}$. This result can be verified by using the Power of a Power Property.