Question
Rewrite \( \sqrt[3]{192} \) in simplest form
Ask by Joseph Turnbull. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \sqrt[3]{192} = 4\sqrt[3]{3} \)
Solution
To simplify ³√192, start by factoring 192 to find a perfect cube factor. Notice that 64 is a perfect cube (since 4³ = 64) and that 64 × 3 = 192. Then:
³√192 = ³√(64 × 3) = ³√64 × ³√3 = 4 × ³√3
Thus, the simplest form is 4³√3.
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Simplify this solution Mind Expander
To simplify \( \sqrt[3]{192} \), we start by factoring 192 into its prime factors. The prime factorization of 192 is \( 2^6 \times 3^1 \). Rewrite the cube root as: \[ \sqrt[3]{192} = \sqrt[3]{2^6 \times 3^1} = \sqrt[3]{(2^6)} \times \sqrt[3]{(3^1)}. \] Now, simplify each term. The cube root of \( 2^6 \) is \( 2^{6/3} = 2^2 = 4 \), and the cube root of \( 3^1 \) is \( 3^{1/3} \). Therefore, we have: \[ \sqrt[3]{192} = 4 \times \sqrt[3]{3}. \] So, the simplest form of \( \sqrt[3]{192} \) is: \[ 4\sqrt[3]{3}. \]
