Simplify and find the value of $\newline$ $(18)^{\frac{1}{3}} \times (768)^{\frac{1}{3}}$

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Algebra 2

Multiplication with rational exponents

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Q. Simplify and find the value of

\newline

(18)^{\frac{1}{3}} \times (768)^{\frac{1}{3}}

Understand the Problem: Understand the problem. We need to simplify the expression $(18)^{1/3}\times(768)^{1/3}$ by finding the cube root of each number and then multiplying the results.
Simplify Each Term: Simplify each term separately. $\newline$ First, we find the cube root of $18$ , which is $(18)^{1/3}$ . $\newline$ Second, we find the cube root of $768$ , which is $(768)^{1/3}$ .
Calculate Cube Root of $18$ : Calculate the cube root of $18$ . $\newline$ The cube root of $18$ is not a whole number, but we can simplify it by looking for factors of $18$ that are perfect cubes. Since $18 = 2 \times 9$ and $9$ is a perfect cube ( $3^2$ ), we can write: $\newline$ $(18)^{1/3} = (2 \times 9)^{1/3}$ $\newline$ $= 2^{1/3} \times 9^{1/3}$ $\newline$ $= 2^{1/3} \times 3$
Calculate Cube Root of $768$ : Calculate the cube root of $768$ . We look for factors of $768$ that are perfect cubes. $768$ can be factored into $2^8 \times 3$ . Since $2^3$ is a perfect cube, we can write: $\newline$ $(768)^{1/3} = (2^8 \times 3)^{1/3}$ $\newline$ $= (2^3)^{8/3} \times 3^{1/3}$ $\newline$ $= 2^{8/3} \times 3^{1/3}$ $\newline$ $= 2^{2\ast4/3} \times 3^{1/3}$ $\newline$ $= 4 \times 2^{4/3} \times 3^{1/3}$
Combine the Results: Combine the results. $\newline$ Now we multiply the simplified cube roots from Step $3$ and Step $4$ : $\newline$ $(2^{1/3} \times 3) \times (4 \times 2^{4/3} \times 3^{1/3})$ $\newline$ = $4 \times 2^{1/3 + 4/3} \times 3 \times 3^{1/3}$ $\newline$ = $4 \times 2^{5/3} \times 3 \times 3^{1/3}$
Simplify Further: Simplify the expression further. $\newline$ Since $2^{5/3}$ is the same as $2^{1+2/3}$ , which is $2 \times 2^{2/3}$ , and $3 \times 3^{1/3}$ is the same as $3^{1+1/3}$ , which is $3^{4/3}$ , we can write: $\newline$ $4 \times 2 \times 2^{2/3} \times 3^{4/3}$ $\newline$ = $8 \times 2^{2/3} \times 3^{4/3}$
Recognize Limitations: Recognize that $2^{\frac{2}{3}}$ and $3^{\frac{4}{3}}$ cannot be simplified further without a calculator. $\newline$ Since we cannot simplify $2^{\frac{2}{3}}$ and $3^{\frac{4}{3}}$ into whole numbers, we leave the expression as is or use a calculator to find the decimal approximation.
Calculate Final Value: Calculate the final value using a calculator (if necessary). $\newline$ If we want to find the decimal approximation of the final value, we would use a calculator to compute $8 \times 2^{2/3} \times 3^{4/3}$ . However, without a calculator, the simplified form of the expression is the final answer.