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Which expression is equivalent to

3root(4)(2^(8)y^(16))*2root(6)(2^(6)y^(12))*root(8)(2^(24)y^(16))" where "y > 0?

$6$ . Which expression is equivalent to $\newline$ $3 \sqrt[4]{2^{8} y^{16}} \cdot 2 \sqrt[6]{2^{6} y^{12}} \cdot \sqrt[8]{2^{24} y^{16}} \text { where } y>0 ?$

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Multiplication with rational exponents

Full solution

Q.

6

. Which expression is equivalent to

\newline

3 \sqrt[4]{2^{8} y^{16}} \cdot 2 \sqrt[6]{2^{6} y^{12}} \cdot \sqrt[8]{2^{24} y^{16}} \text { where } y>0 ?

Simplify Terms: Step $1$ : Simplify each term separately. $\newline$ We start by simplifying each root term: $\newline$ - For the first term, $3\sqrt[4]{2^8y^{16}}$ , simplify the powers under the root: $\newline$ $2^8 = 2^{2 \times 4}$ and $y^{16} = y^{4 \times 4}$ , $\newline$ so $3\sqrt[4]{2^8y^{16}} = 3 \times 2^2y^4 = 12y^4$ .
Simplify Second Term: Step $2$ : Simplify the second term. $\newline$ - For the second term, $2\sqrt[6]{2^6y^{12}}$ , simplify the powers under the root: $\newline$ $2^6 = 2^{1 \times 6}$ and $y^{12} = y^{2 \times 6}$ , $\newline$ so $2\sqrt[6]{2^6y^{12}} = 2 \times 2^1y^2 = 4y^2$ .
Simplify Third Term: Step $3$ : Simplify the third term. $\newline$ - For the third term, $\sqrt[8]{2^{24}y^{16}}$ , simplify the powers under the root: $\newline$ $2^{24} = 2^{3 \times 8}$ and $y^{16} = y^{2 \times 8}$ , $\newline$ so $\sqrt[8]{2^{24}y^{16}} = 2^3y^2 = 8y^2$ .
Multiply Simplified Terms: Step $4$ : Multiply all simplified terms together. $\newline$ - Multiply $12y^4$ , $4y^2$ , and $8y^2$ together: $\newline$ $12y^4 \times 4y^2 \times 8y^2 = 12 \times 4 \times 8 \times y^{4+2+2} = 384y^8$ .

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