Select all the expressions that are equivalent to $3^6 \times 9^6$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{27^{-6}}$ $\newline$ (B) $27^{12}$ $\newline$ (C) $\frac{1}{27^6}$ $\newline$ (D) $27^{36}$

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Algebra 2

Domain and range of quadratic functions: equations

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Q. Select all the expressions that are equivalent to

3^6 \times 9^6

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{27^{-6}}

\newline

(B)

27^{12}

\newline

(C)

\frac{1}{27^6}

\newline

(D)

27^{36}

Recognize $9$ as Power of $3$ : We need to simplify the expression $3^6 \times 9^6$ to find equivalent expressions. $\newline$ First, we recognize that $9$ is a power of $3$ , specifically $9 = 3^2$ . $\newline$ So, we can rewrite $9^6$ as $(3^2)^6$ .
Apply Power of a Power Rule: Now we apply the power of a power rule, which states that $(a^b)^c = a^{(b*c)}$ . So, $(3^2)^6$ becomes $3^{(2*6)}$ or $3^{12}$ .
Multiply Exponents of Same Base: Next, we multiply the exponents of the same base in the original expression. $\newline$ So, $3^6 \times 3^{12}$ becomes $3^{6+12}$ or $3^{18}$ .
Compare to Choices: Now we have simplified the original expression to $3^{18}$ . We will compare this to each of the choices to see which are equivalent.
Simplify Choice (A): For choice (A) $\frac{1}{27^{-6}}$ , we recognize that $27$ is $3^3$ . So, $\frac{1}{27^{-6}}$ is the same as $\frac{1}{(3^3)^{-6}}$ . Applying the negative exponent rule, which states that $\frac{1}{a^{-b}} = a^b$ , we get $(3^3)^6$ .
Simplify Choice (B): Applying the power of a power rule again, $(3^3)^6$ becomes $3^{(3*6)}$ or $3^{18}$ . So, choice (A) is equivalent to $3^{18}$ .
Simplify Choice (C): For choice (B) $27^{12}$ , we again recognize that $27$ is $3^3$ . So, $27^{12}$ is the same as $(3^3)^{12}$ . Applying the power of a power rule, $(3^3)^{12}$ becomes $3^{(3*12)}$ or $3^{36}$ . This is not equivalent to $3^{18}$ .
Simplify Choice (D): For choice (C) $1/27^6$ , we have $1$ divided by $27$ raised to the $6$ th power. $\newline$ As before, $27$ is $3^3$ , so $1/27^6$ is the same as $1/(3^3)^6$ . $\newline$ This simplifies to $1/3^{18}$ , which is the reciprocal of $3^{18}$ , not equivalent to $3^{18}$ .
Simplify Choice (D): For choice (C) $\frac{1}{27^6}$ , we have $1$ divided by $27$ raised to the $6$ th power. $\newline$ As before, $27$ is $3^3$ , so $\frac{1}{27^6}$ is the same as $\frac{1}{(3^3)^6}$ . $\newline$ This simplifies to $\frac{1}{3^{18}}$ , which is the reciprocal of $3^{18}$ , not equivalent to $3^{18}$ .For choice (D) $1$ $1$ , we have $27$ raised to the $1$ $3$ th power. $\newline$ Since $27$ is $3^3$ , $1$ $1$ is the same as $1$ $7$ . $\newline$ Applying the power of a power rule, $1$ $7$ becomes $1$ $9$ or $27$ $0$ . $\newline$ This is not equivalent to $3^{18}$ .