Solve for a ((x^(1/3))^2)/( cube root of x^2)=x^a

$\frac{{(x^{\frac{1}{3}})}^{2}}{\sqrt[3]{x^{2}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{{(x^{\frac{1}{3}})}^{2}}{\sqrt[3]{x^{2}}}$ .

$x^{a} = \frac{{(x^{\frac{1}{3}})}^{2}}{\sqrt[3]{x^{2}}}$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (x^{a}) = \ln (\frac{{(x^{\frac{1}{3}})}^{2}}{\sqrt[3]{x^{2}}})$

Step 3

Expand $\ln (x^{a})$ by moving $a$ outside the logarithm.

$a \ln (x) = \ln (\frac{{(x^{\frac{1}{3}})}^{2}}{\sqrt[3]{x^{2}}})$

Step 4

Simplify the right side.

Simplify $\ln (\frac{{(x^{\frac{1}{3}})}^{2}}{\sqrt[3]{x^{2}}})$ .

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$a \ln (x) = \ln (\frac{x^{\frac{1}{3} \cdot 2}}{\sqrt[3]{x^{2}}})$

Combine $\frac{1}{3}$ and $2$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}}}{\sqrt[3]{x^{2}}})$

Multiply $\frac{x^{\frac{2}{3}}}{\sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}}}{\sqrt[3]{x^{2}}} \cdot \frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}})$

Combine and simplify the denominator.

Multiply $\frac{x^{\frac{2}{3}}}{\sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{\sqrt[3]{x^{2}} {\sqrt[3]{x^{2}}}^{2}})$

Raise $\sqrt[3]{x^{2}}$ to the power of $1$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1} {\sqrt[3]{x^{2}}}^{2}})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1 + 2}})$

Add $1$ and $2$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{3}})$

Rewrite ${\sqrt[3]{x^{2}}}^{3}$ as $x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{{(x^{\frac{2}{3}})}^{3}})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{3} \cdot 3}})$

Combine $\frac{2}{3}$ and $3$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2 \cdot 3}{3}}})$

Multiply $2$ by $3$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{6}{3}}})$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3}}})$

Cancel the common factors.

Factor $3$ out of $3$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 (1)}}})$

Cancel the common factor.

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 \cdot 1}}})$

Rewrite the expression.

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{1}}})$

Divide $2$ by $1$ .

$a \ln (x) = \ln (\frac{x^{\frac{2}{3}} {\sqrt[3]{x^{2}}}^{2}}{x^{2}})$

Move $x^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$a \ln (x) = \ln (\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{- \frac{2}{3}}})$

Multiply $x^{2}$ by $x^{- \frac{2}{3}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a \ln (x) = \ln (\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{2 - \frac{2}{3}}})$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a \ln (x) = \ln (\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{2 \cdot \frac{3}{3} - \frac{2}{3}}})$

Combine $2$ and $\frac{3}{3}$ .

$a \ln (x) = \ln (\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2 \cdot 3}{3} - \frac{2}{3}}})$

Combine the numerators over the common denominator.

$a \ln (x) = \ln (\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2 \cdot 3 - 2}{3}}})$

Simplify the numerator.

Multiply $2$ by $3$ .

$a \ln (x) = \ln (\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{\frac{6 - 2}{3}}})$

Subtract $2$ from $6$ .

$a \ln (x) = \ln (\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{\frac{4}{3}}})$

Simplify the numerator.

Rewrite ${\sqrt[3]{x^{2}}}^{2}$ as ${({(x^{2})}^{2})}^{\frac{1}{3}}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{{(x^{2})}^{2}}}{x^{\frac{4}{3}}})$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{2 \cdot 2}}}{x^{\frac{4}{3}}})$

Multiply $2$ by $2$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{4}}}{x^{\frac{4}{3}}})$

Factor out $x^{3}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{3} x}}{x^{\frac{4}{3}}})$

Pull terms out from under the radical.

$a \ln (x) = \ln (\frac{x \sqrt[3]{x}}{x^{\frac{4}{3}}})$

Move $x^{1}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{4}{3}} x^{- 1}})$

Multiply $x^{\frac{4}{3}}$ by $x^{- 1}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{4}{3} - 1}})$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}})$

Combine $- 1$ and $\frac{3}{3}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}})$

Combine the numerators over the common denominator.

$a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{4 - 1 \cdot 3}{3}}})$

Simplify the numerator.

Multiply $- 1$ by $3$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{4 - 3}{3}}})$

Subtract $3$ from $4$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{1}{3}}})$

Step 5

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{1}{3}}})$ by $\ln (x)$ and simplify.

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{1}{3}}})$ by $\ln (x)$ .

$\frac{a \ln (x)}{\ln (x)} = \frac{\ln (\frac{\sqrt[3]{x}}{x^{\frac{1}{3}}})}{\ln (x)}$

Simplify the left side.

Cancel the common factor of $\ln (x)$ .

Cancel the common factor.

$\frac{a \ln (x)}{\ln (x)} = \frac{\ln (\frac{\sqrt[3]{x}}{x^{\frac{1}{3}}})}{\ln (x)}$

Divide $a$ by $1$ .

$a = \frac{\ln (\frac{\sqrt[3]{x}}{x^{\frac{1}{3}}})}{\ln (x)}$

Solve for a ((x^(1/3))^2)/( cube root of x^2)=x^a

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