Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
negative exponents
multiplying with like bases
choose two correct answers that are equal to $5^3 \cdot 5^{-7}$.
options: $5^{-4}$, $\frac{1}{5^{-1}}$, $\frac{1}{5^4}$, $5^4$, $-\frac{1}{625}$

Explanation:

Step1: Apply exponent rule for multiplication

When multiplying exponents with the same base, we use the rule \( a^m \cdot a^n = a^{m + n} \). For \( 5^3 \cdot 5^{-7} \), we add the exponents: \( 3 + (-7)=3 - 7=-4 \). So \( 5^3 \cdot 5^{-7}=5^{-4} \).

Step2: Apply negative exponent rule

The negative exponent rule states that \( a^{-n}=\frac{1}{a^n} \). So \( 5^{-4}=\frac{1}{5^4} \). Also, let's check other options:

• \( \frac{1}{5^{-1}} = 5^1 = 5

eq5^{-4} \)

• \( 5^4

eq5^{-4} \)

• \( -\frac{1}{625}

eq\frac{1}{5^4} \) (since \( 5^4 = 625 \), \( \frac{1}{5^4}=\frac{1}{625} \))

Answer:

A. \( 5^{-4} \), B. \( \frac{1}{5^4} \) (assuming the options are labeled as top - left: \( \frac{1}{5^{-1}} \), top - right: \( 5^{-4} \), middle - left: \( 5^4 \), middle - right: \( \frac{1}{5^4} \), bottom: \( -\frac{1}{625} \))