Question

What is the following quotient? StartFraction 6 minus 3 (RootIndex 3 StartRoot 6 EndRoot) Over RootIndex 3 StartRoot 9 EndRoot EndFraction.

Answer 1

Exponent properties help us to simplify the powers of expressions.  The quotient of the given expression $\dfrac{6 - 3(\sqrt[3]{6})}{\sqrt[3]{9}}$ is (2∛3 - ∛18).

What are the basic exponent properties?

${a^m} \cdot {a^n} = a^{(m+n)}\\\\\dfrac{a^m}{a^n} = a^{(m-n)}\\\\\sqrt[m]{a^n} = a^{\frac{n}{m}}\\\\(a^m)^n = a^{m\times n}\\\\(m\times n)^a = m^a\times n^a$

Given to us

$\dfrac{6 - 3(\sqrt[3]{6})}{\sqrt[3]{9}}$

We will solve the problem using the basic exponential properties,

$\dfrac{6 - 3(\sqrt[3]{6})}{\sqrt[3]{9}}\\\\ = \dfrac{6}{\sqrt[3]{9}} - \dfrac{3(\sqrt[3]{6})}{\sqrt[3]{9}}\\\\ = (6\cdot 3^{-\frac{2}{3}}) - [3 \cdot (2 \cdot 3)^{-\frac{2}{3}}3^{-\frac{2}{3}}]\\\\= (2 \cdot 3 \cdot 3^{-\frac{2}{3}}) - [3 \cdot 2^{-\frac{2}{3}} \cdot 3^{-\frac{2}{3}}3^{-\frac{2}{3}}]$

$= [2 \cdot 3^{(1-\frac{2}{3})}] - [2^{\frac{1}{3}}\cdot 3^{(1+\frac{1}{3} - \frac{2}{3})}]\\\\= [2 \cdot 3^{(\frac{1}{3})}] - [2^{\frac{1}{3}}\cdot 3^{(\frac{2}{3})}]\\\\= 2\sqrt[3]{3} - \sqrt[3]{2}\sqrt[3]{9}\\\\=2\sqrt[3]{3} - \sqrt[3]{18}$

Hence, the quotient of the given expression $\dfrac{6 - 3(\sqrt[3]{6})}{\sqrt[3]{9}}$ is (2∛3 - ∛18).

Learn more about Exponents:

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