Question

select all that are equal to $5^{3} \bullet 5^{-7}$
$\frac{1}{5^{-4}}$
$\frac{1}{5^{4}}$
$5^{-4}$
$5^{4}$
$\frac{1}{-625}$

Explanation:

Step1: Apply exponent product rule

When multiplying same bases, add exponents: $5^{3} \cdot 5^{-7} = 5^{3 + (-7)} = 5^{-4}$

Step2: Rewrite negative exponent

A negative exponent means reciprocal: $5^{-4} = \frac{1}{5^{4}}$

Step3: Evaluate the numerical value

Calculate $5^4 = 5 \times 5 \times 5 \times 5 = 625$, so $\frac{1}{5^4} = \frac{1}{625}$

Step4: Analyze each option

• $\frac{1}{5^{-4}} = 5^4$, not equal
• $\frac{1}{5^4}$ matches the rewritten form
• $5^{-4}$ matches the simplified exponent form
• $5^4$ is the reciprocal, not equal
• $\frac{1}{-625}$ is negative, not equal

Answer:

$\frac{1}{5^{4}}$, $5^{-4}$