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6^((1)/(6))*2^((5)/(6))*3^((7)/(6))
Which of the following is equivalent to the given expression?
Choose 1 answer:
(A)
6root(3)(3)
(B)
2^((5)/(36))*3^((7)/(36))
(C)
36^((35)/(216))
(D)
root(3)(6^(13))

$6^{\frac{1}{6}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{7}{6}}$ $\newline$ Which of the following is equivalent to the given expression? $\newline$ Choose $1$ answer: $\newline$ (A) $6 \sqrt[3]{3}$ $\newline$ (B) $2^{\frac{5}{36}} \cdot 3^{\frac{7}{36}}$ $\newline$ (C) $36^{\frac{35}{216}}$ $\newline$ (D) $\sqrt[3]{6^{13}}$

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Compare linear and exponential growth

Full solution

Q.

6^{\frac{1}{6}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{7}{6}}

\newline

Which of the following is equivalent to the given expression?

\newline

Choose

1

answer:

\newline

(A)

6 \sqrt[3]{3}

\newline

(B)

2^{\frac{5}{36}} \cdot 3^{\frac{7}{36}}

\newline

(C)

36^{\frac{35}{216}}

\newline

(D)

\sqrt[3]{6^{13}}

Combine terms with same base: Combine the terms with the same base. $\newline$ We notice that $6$ can be written as $2 \times 3$ , so we can combine the terms with the base of $2$ and $3$ . $\newline$ $6^{(1/6)}\times2^{(5/6)}\times3^{(7/6)} = (2 \times 3)^{(1/6)}\times2^{(5/6)}\times3^{(7/6)}$
Apply exponent to factors: Apply the exponent to the factors of $6$ . $\newline$ $(2 \times 3)^{\frac{1}{6}} = 2^{\frac{1}{6}} \times 3^{\frac{1}{6}}$ $\newline$ Now we have all terms with base $2$ and $3$ , and we can combine them.
Combine exponents for base terms: Combine the exponents for the base $2$ and base $3$ terms. $\newline$ $2^{(1)/(6)} \times 2^{(5)/(6)} = 2^{(1)/(6) + (5)/(6)} = 2^1 = 2$ $\newline$ $3^{(1)/(6)} \times 3^{(7)/(6)} = 3^{(1)/(6) + (7)/(6)} = 3^{(8)/(6)} = 3^{4/3}$
Multiply simplified terms: Multiply the simplified terms. $2 \times 3^{\frac{4}{3}} = 2 \times \sqrt[3]{3^4} = 2 \times \sqrt[3]{81}$
Simplify cube root: Simplify the cube root. $\sqrt[3]{81} = 3^{4^{1/3}} = 3^{4/3} = 3^{1*4/3} = 3^{4/3} = 3 \times 3 = 9$
Multiply remaining terms: Multiply the remaining terms. $2 \times 9 = 18$
Check answer choices: Check the answer choices to see which one is equivalent to $18$ .
(A) $6\sqrt[3]{3}$ is not $18$ .
(B) $2^{\frac{5}{36}}\times3^{\frac{7}{36}}$ is not $18$ .
(C) $36^{\frac{35}{216}}$ is not $18$ .
(D) $\sqrt[3]{6^{13}}$ is not $18$ .
None of the answer choices are equivalent to $18$ , which means there might be a mistake in our calculations.

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