Question

which expression is equivalent to \\(\frac{12^3}{2^3}\\)?
options: 6, \\(\frac{1}{6^{-3}}\\), \\(6^0\\), \\(\frac{6^6}{6^2}\\)

Explanation:

Step1: Rewrite numerator as product

$12^3 = (6 \times 2)^3 = 6^3 \times 2^3$

Step2: Substitute into original expression

$\frac{6^3 \times 2^3}{2^3}$

Step3: Cancel common terms

$\frac{6^3 \times \cancel{2^3}}{\cancel{2^3}} = 6^3$

Step4: Simplify the option $\frac{1}{6^{-3}}$

Using $a^{-n} = \frac{1}{a^n}$, so $\frac{1}{6^{-3}} = 6^3$

Step5: Verify other options

• $6

eq 6^3$

• $6^0 = 1

eq 6^3$

• $\frac{6^6}{6^2} = 6^{6-2} = 6^4

eq 6^3$

Answer:

$\boldsymbol{\frac{1}{6^{-3}}}$