Which expression is equivalent to $3^6 \times 6^6$ ? $\newline$ Choices: $\newline$ (A) $\frac{1}{18^6}$ $\newline$ (B) $18^{12}$ $\newline$ (C) $18^6$ $\newline$ (D) $18^{36}$

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Grade 8

Identify equivalent expressions involving exponents I

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Q. Which expression is equivalent to

3^6 \times 6^6

\newline

Choices:

\newline

(A)

\frac{1}{18^6}

\newline

(B)

18^{12}

\newline

(C)

18^6

\newline

(D)

18^{36}

Identify Bases and Exponents: Identify the bases and the exponents in the expression $3^6 \times 6^6$ . We have two terms, $3^6$ and $6^6$ , with bases $3$ and $6$ , and both raised to the exponent $6$ .
Prime Factors of $6$ : Recognize that $6$ can be written as a product of its prime factors. $\newline$ The number $6$ is equal to $2 \times 3$ . Therefore, $6^6$ can be rewritten as $(2 \times 3)^6$ .
Power of a Product Rule: Apply the power of a product rule to $(2 \times 3)^6$ . According to the power of a product rule, $(ab)^n = a^n \times b^n$ . So, $(2 \times 3)^6 = 2^6 \times 3^6$ .
Substitute and Combine Terms: Substitute $6^6$ with $2^6 \times 3^6$ in the original expression. $\newline$ Now we have $3^6 \times 6^6 = 3^6 \times (2^6 \times 3^6)$ .
Combine Like Terms: Combine like terms by adding the exponents of the same base. $\newline$ We have two terms with the base $3$ , so we add their exponents: $3^6 \times 3^6 = 3^{6+6} = 3^{12}$ .
Multiply Remaining Terms: Multiply the remaining terms. $\newline$ Now we have $3^{12} \times 2^{6}$ . Since these terms do not have the same base, we cannot combine them further.
Power of a Power Rule: Recognize that $3^{12}$ can be written as $(3^6)^2$ . Using the power of a power rule, $(a^n)^m = a^{n*m}$ , we have $(3^6)^2 = 3^{6*2} = 3^{12}$ .
Combine Expressions: Combine the expressions $(3^6)^2$ and $2^6$ . Now we have $(3^6)^2 \times 2^6 = 3^{12} \times 2^6$ .
Rearrange Terms: Recognize that $3^{12} \times 2^{6}$ can be written as $(3^{6} \times 2^{6}) \times 3^{6}$ . We can rearrange the terms to group them as $(3^{6} \times 2^{6}) \times 3^{6}$ .
Substitute Back Expression: Substitute back the expression for $6^6$ . We know that $6^6 = 2^6 \times 3^6$ , so we can write $(3^6 \times 2^6) \times 3^6$ as $6^6 \times 3^6$ .
Full Circle to Original: Recognize that we have come full circle to the original expression. $\newline$ We have shown that $3^6 \times 6^6 = 6^6 \times 3^6$ , which is the same as the original expression.
Simplify the Expression: Simplify the expression $6^6 \times 3^6$ . $\newline$ Since $6^6 = (2 \times 3)^6$ , we can write $6^6 \times 3^6$ as $(2 \times 3)^6 \times 3^6 = 2^6 \times 3^{12}.$
Correct Mistake: Recognize that we have made a mistake in the previous steps. $\newline$ We have incorrectly circled back to the original expression without simplifying it correctly. We need to correct this mistake.