Question

which expressions are equivalent to $5^{12} \cdot 5^{8}$? choose 2 answers: a $25^{20}$ b $(25^{5})^{4}$ c $(5^{3} \cdot 5^{2})^{4}$ d $(5^{5})^{4}$

Explanation:

Step1: Simplify the original expression

Use exponent rule $a^m \cdot a^n = a^{m+n}$:
$5^{12} \cdot 5^8 = 5^{12+8} = 5^{20}$

Step2: Evaluate Option A

Rewrite $25^{20}$ as $(5^2)^{20}$, use rule $(a^m)^n = a^{m \cdot n}$:
$(5^2)^{20} = 5^{40}
eq 5^{20}$

Step3: Evaluate Option B

Rewrite $25^5$ as $(5^2)^5$, then apply exponent rule:
$(25^5)^4 = ((5^2)^5)^4 = 5^{2 \cdot 5 \cdot 4} = 5^{40}
eq 5^{20}$

Step4: Evaluate Option C

First simplify inside the parentheses:
$5^3 \cdot 5^2 = 5^{3+2} = 5^5$
Then apply exponent rule $(a^m)^n = a^{m \cdot n}$:
$(5^5)^4 = 5^{5 \cdot 4} = 5^{20}$

Step5: Evaluate Option D

Apply exponent rule $(a^m)^n = a^{m \cdot n}$:
$(5^5)^4 = 5^{5 \cdot 4} = 5^{20}$

Answer:

C. $(5^{3} \cdot 5^{2})^{4}$, D. $(5^{5})^{4}$