8. $(64)^{\frac{1}{3}} \times (128)^{\frac{2}{7}} =$

$\bigcirc$ 8
$\bigcirc$ 16
$\bigcirc$ 42
$\bigcirc$ 128
$\bigcirc$ 256

Question

8. $(64)^{\frac{1}{3}} \times (128)^{\frac{2}{7}} =$

$\bigcirc$ 8
$\bigcirc$ 16
$\bigcirc$ 42
$\bigcirc$ 128
$\bigcirc$ 256

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Exponent Rules", "Product of Powers" ], "new_concepts": [ "Rational Exponents", "Prime Factorization" ], "current_concepts": [ "Exponent Rules", "Product of Powers", "Rational Exponents", "Prime Factorization" ] } ## Express bases as powers of 2 Using the *Exponent Rules* knowledge point $$ \begin{aligned} 64 &= 2^6 \ 128 &= 2^7 \end{aligned} $$ ## Substitute and apply power of a power rule Using the *Exponent Rules* knowledge point $$ \begin{aligned} (64)^{\frac{1}{3}} &= (2^6)^{\frac{1}{3}} = 2^{6 \times \frac{1}{3}} = 2^2 \ (128)^{\frac{2}{7}} &= (2^7)^{\frac{2}{7}} = 2^{7 \times \frac{2}{7}} = 2^2 \end{aligned} $$ ## Multiply the simplified terms Using the *Product of Powers* knowledge point $$ 2^2 \times 2^2 = 2^{2+2} = 2^4 = 16 $$

8

16

42

128

256

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponent Rules" ] }

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QUESTION IMAGE

Question

Explanation:

Express bases as powers of 2

Substitute and apply power of a power rule

Multiply the simplified terms

Answer: