Question

1. factual recall

use properties of exponents to simplify.

a) $x^4 \cdot x^3 =$

b) $a^0b^{-3} =$

c) $(w^2)^3 =$

1. prove, show, justify

show that
$((2m^2)^3)^2 = 64m^{12}$

Explanation:

Part 1: Factual Recall (Simplify using exponent properties)

a) \( x^4 \cdot x^3 \)

Step1: Use product of exponents rule (\( a^m \cdot a^n = a^{m + n} \))

\( x^4 \cdot x^3 = x^{4 + 3} \)

Step2: Simplify the exponent

\( x^{4 + 3} = x^7 \)

Step1: Use zero - exponent rule (\( a^0 = 1, a

eq0 \))
\( a^0 = 1 \), so the expression becomes \( 1\times b^{-3} \)

Step2: Use negative - exponent rule (\( a^{-n}=\frac{1}{a^n} \))

\( b^{-3}=\frac{1}{b^3} \), and \( 1\times\frac{1}{b^3}=\frac{1}{b^3} \)

Step1: Use power of a power rule (\( (a^m)^n=a^{m\times n} \))

\( (w^2)^3 = w^{2\times3} \)

Step2: Simplify the exponent

\( w^{2\times3}=w^6 \)

Answer:

\( x^7 \)

b) \( a^0 b^{-3} \)