Select the equivalent expression. $\newline$ $(3^{3}\cdot6^{6})^{-3}=$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{3^{9}\cdot6^{18}}$ $\newline$ (B) $\frac{6^{18}}{3^{9}}$ $\newline$ (C) $\frac{3^{9}}{6^{18}}$

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Math Problems

Grade 8

Evaluate expressions using properties of exponents

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Q. Select the equivalent expression.

\newline

(3^{3}\cdot6^{6})^{-3}=

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{3^{9}\cdot6^{18}}

\newline

(B)

\frac{6^{18}}{3^{9}}

\newline

(C)

\frac{3^{9}}{6^{18}}

Understand Problem: Understand the problem and apply the power of a product rule. $\newline$ The problem asks us to find an equivalent expression for $(3^{3}\cdot6^{6})^{-3}$ . According to the power of a product rule, $(ab)^{n} = a^{n} \cdot b^{n}$ , we can apply the negative exponent to both terms inside the parentheses.
Apply Product Rule: Apply the negative exponent to both $3^3$ and $6^6$ . $\newline$ $(3^{3}*6^{6})^{(-3)} = (3^3)^{(-3)} * (6^6)^{(-3)}$
Apply Negative Exponent: Simplify the expression using the power of a power rule. $\newline$ According to the power of a power rule, $(a^m)^n = a^{m*n}$ , we multiply the exponents. $\newline$ $(3^3)^{-3} \times (6^6)^{-3} = 3^{3*(-3)} \times 6^{6*(-3)}$
Simplify Expression: Perform the multiplication of the exponents. $3^{3*(-3)} \times 6^{6*(-3)} = 3^{-9} \times 6^{-18}$
Perform Exponent Multiplication: Convert the negative exponents to positive exponents by taking the reciprocal. $\newline$ According to the negative exponent rule, $a^{-n} = \frac{1}{a^n}$ , we can rewrite the expression with positive exponents in the denominator. $\newline$ $3^{-9} \times 6^{-18} = \frac{1}{3^9} \times \frac{1}{6^{18}}$
Convert Negative Exponents: Combine the fractions. $\newline$ Since we are multiplying fractions, we multiply the numerators and the denominators separately. $\newline$ $\frac{1}{3^9} \times \frac{1}{6^{18}} = \frac{1}{3^9 \times 6^{18}}$
Combine Fractions: Identify the correct answer from the given options. $\newline$ The expression we have found is $\frac{1}{3^9 \times 6^{18}}$ , which matches option $(A)$ .