Solve for w ( fourth root of 7w-5)/2 = fourth root of w-2

$\frac{\sqrt[4]{7 w - 5}}{2} = \sqrt[4]{w - 2}$

Step 1

Solve for $\sqrt[4]{7 w - 5}$ .

Multiply both sides of the equation by $2$ .

$2 \frac{\sqrt[4]{7 w - 5}}{2} = 2 \sqrt[4]{w - 2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{\sqrt[4]{7 w - 5}}{2} = 2 \sqrt[4]{w - 2}$

Rewrite the expression.

$\sqrt[4]{7 w - 5} = 2 \sqrt[4]{w - 2}$

Step 2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{7 w - 5}}^{4} = {(2 \sqrt[4]{w - 2})}^{4}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{7 w - 5}$ as ${(7 w - 5)}^{\frac{1}{4}}$ .

${({(7 w - 5)}^{\frac{1}{4}})}^{4} = {(2 \sqrt[4]{w - 2})}^{4}$

Simplify the left side.

Simplify ${({(7 w - 5)}^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${({(7 w - 5)}^{\frac{1}{4}})}^{4}$ .

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

${(7 w - 5)}^{\frac{1}{4} \cdot 4} = {(2 \sqrt[4]{w - 2})}^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(7 w - 5)}^{\frac{1}{4} \cdot 4} = {(2 \sqrt[4]{w - 2})}^{4}$

Rewrite the expression.

${(7 w - 5)}^{1} = {(2 \sqrt[4]{w - 2})}^{4}$

Simplify.

$7 w - 5 = {(2 \sqrt[4]{w - 2})}^{4}$

Simplify the right side.

Simplify ${(2 \sqrt[4]{w - 2})}^{4}$ .

Simplify the expression.

Apply the product rule to $2 \sqrt[4]{w - 2}$ .

$7 w - 5 = 2^{4} {\sqrt[4]{w - 2}}^{4}$

Raise $2$ to the power of $4$ .

$7 w - 5 = 16 {\sqrt[4]{w - 2}}^{4}$

Rewrite ${\sqrt[4]{w - 2}}^{4}$ as $w - 2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{w - 2}$ as ${(w - 2)}^{\frac{1}{4}}$ .

$7 w - 5 = 16 {({(w - 2)}^{\frac{1}{4}})}^{4}$

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

$7 w - 5 = 16 {(w - 2)}^{\frac{1}{4} \cdot 4}$

Combine $\frac{1}{4}$ and $4$ .

$7 w - 5 = 16 {(w - 2)}^{\frac{4}{4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$7 w - 5 = 16 {(w - 2)}^{\frac{4}{4}}$

Rewrite the expression.

$7 w - 5 = 16 {(w - 2)}^{1}$

Simplify.

$7 w - 5 = 16 (w - 2)$

Apply the distributive property.

$7 w - 5 = 16 w + 16 \cdot - 2$

Multiply $16$ by $- 2$ .

$7 w - 5 = 16 w - 32$

Step 4

Solve for $w$ .

Move all terms containing $w$ to the left side of the equation.

Subtract $16 w$ from both sides of the equation.

$7 w - 5 - 16 w = - 32$

Subtract $16 w$ from $7 w$ .

$- 9 w - 5 = - 32$

Move all terms not containing $w$ to the right side of the equation.

Add $5$ to both sides of the equation.

$- 9 w = - 32 + 5$

Add $- 32$ and $5$ .

$- 9 w = - 27$

Divide each term in $- 9 w = - 27$ by $- 9$ and simplify.

Divide each term in $- 9 w = - 27$ by $- 9$ .

$\frac{- 9 w}{- 9} = \frac{- 27}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 w}{- 9} = \frac{- 27}{- 9}$

Divide $w$ by $1$ .

$w = \frac{- 27}{- 9}$

Simplify the right side.

Divide $- 27$ by $- 9$ .

$w = 3$

Solve for w ( fourth root of 7w-5)/2 = fourth root of w-2

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