Question

how can you use patterns to simplify the algebraic expression $x^3 \cdot x^6$? what property can you use to verify your answer? explain your reasoning. by the definition of an exponent, the first and second factors of $x^3 \cdot x^6$ can be written respectively as the product of $\boldsymbol{\
abla}$ factors of $\boldsymbol{\
abla}$ and the product of $\boldsymbol{\
abla}$ factors of $\boldsymbol{\
abla}$ in all, there are thus $\boldsymbol{\square}$ factors of $\boldsymbol{\
abla}$ which can be written as the exponential expression $\boldsymbol{\square}$. this result can be verified by using the $\boldsymbol{\
abla}$ property. (use integers or fractions for any numbers in the expressions.)

Explanation:

Step1: Expand exponents by definition

$x^3 = x \cdot x \cdot x$ (3 factors of $x$), $x^6 = x \cdot x \cdot x \cdot x \cdot x \cdot x$ (6 factors of $x$)

Step2: Count total factors of $x$

$3 + 6 = 9$ total factors of $x$

Step3: Rewrite as exponential expression

9 factors of $x$ is $x^9$

Step4: Identify verifying property

This follows the product rule for exponents.

Answer:

By the definition of an exponent, the first and second factors of $x^3 \cdot x^6$ can be written respectively as the product of $\boldsymbol{3}$ factors of $\boldsymbol{x}$ and the product of $\boldsymbol{6}$ factors of $\boldsymbol{x}$. In all, there are thus $\boldsymbol{9}$ factors of $\boldsymbol{x}$ which can be written as the exponential expression $\boldsymbol{x^9}$. This result can be verified by using the Product of Powers Property.