Find All Complex Solutions cube root of 6/7=( cube root of 6)/( cube root of 7)

$\sqrt[3]{\frac{6}{7}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 1

Move all terms not containing a variable to the right side of the equation.

Subtract $\sqrt[3]{\frac{6}{7}}$ from both sides of the equation.

$0 = \frac{\sqrt[3]{6}}{\sqrt[3]{7}} - \sqrt[3]{\frac{6}{7}}$

Simplify each term.

Multiply $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$0 = \frac{\sqrt[3]{6}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}} - \sqrt[3]{\frac{6}{7}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}} - \sqrt[3]{\frac{6}{7}}$

Raise $\sqrt[3]{7}$ to the power of $1$ .

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}} - \sqrt[3]{\frac{6}{7}}$

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

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}} - \sqrt[3]{\frac{6}{7}}$

Add $1$ and $2$ .

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}} - \sqrt[3]{\frac{6}{7}}$

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}} - \sqrt[3]{\frac{6}{7}}$

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

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}} - \sqrt[3]{\frac{6}{7}}$

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

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}} - \sqrt[3]{\frac{6}{7}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}} - \sqrt[3]{\frac{6}{7}}$

Rewrite the expression.

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{1}} - \sqrt[3]{\frac{6}{7}}$

Evaluate the exponent.

$0 = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7} - \sqrt[3]{\frac{6}{7}}$

Simplify the numerator.

Rewrite ${\sqrt[3]{7}}^{2}$ as ${(7^{2})}^{\frac{1}{3}}$ .

$0 = \frac{\sqrt[3]{6} \sqrt[3]{7^{2}}}{7} - \sqrt[3]{\frac{6}{7}}$

Raise $7$ to the power of $2$ .

$0 = \frac{\sqrt[3]{6} \sqrt[3]{49}}{7} - \sqrt[3]{\frac{6}{7}}$

Simplify the numerator.

Combine using the product rule for radicals.

$0 = \frac{\sqrt[3]{6 \cdot 49}}{7} - \sqrt[3]{\frac{6}{7}}$

Multiply $6$ by $49$ .

$0 = \frac{\sqrt[3]{294}}{7} - \sqrt[3]{\frac{6}{7}}$

Rewrite $\sqrt[3]{\frac{6}{7}}$ as $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ .

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Multiply $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$0 = \frac{\sqrt[3]{294}}{7} - (\frac{\sqrt[3]{6}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Raise $\sqrt[3]{7}$ to the power of $1$ .

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}}$

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

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}}$

Add $1$ and $2$ .

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}}$

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}}$

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

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}}$

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

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Rewrite the expression.

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{1}}$

Evaluate the exponent.

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7}$

Simplify the numerator.

Rewrite ${\sqrt[3]{7}}^{2}$ as ${(7^{2})}^{\frac{1}{3}}$ .

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} \sqrt[3]{7^{2}}}{7}$

Raise $7$ to the power of $2$ .

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6} \sqrt[3]{49}}{7}$

Simplify the numerator.

Combine using the product rule for radicals.

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{6 \cdot 49}}{7}$

Multiply $6$ by $49$ .

$0 = \frac{\sqrt[3]{294}}{7} - \frac{\sqrt[3]{294}}{7}$

Combine the numerators over the common denominator.

$0 = \frac{\sqrt[3]{294} - \sqrt[3]{294}}{7}$

Subtract $\sqrt[3]{294}$ from $\sqrt[3]{294}$ .

$0 = \frac{0}{7}$

Divide $0$ by $7$ .

$0 = 0$

Step 2

Since $0 = 0$ , the equation will always be true.

Always true

Find All Complex Solutions cube root of 6/7=( cube root of 6)/( cube root of 7)

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