Solve by Factoring (x-5)^(2/5)=9

${(x - 5)}^{\frac{2}{5}} = 9$

Step 1

Subtract $9$ from both sides of the equation.

${(x - 5)}^{\frac{2}{5}} - 9 = 0$

Step 2

Rewrite ${(x - 5)}^{\frac{2}{5}}$ as ${({(x - 5)}^{\frac{1}{5}})}^{2}$ .

${({(x - 5)}^{\frac{1}{5}})}^{2} - 9 = 0$

Step 3

Rewrite $9$ as $3^{2}$ .

${({(x - 5)}^{\frac{1}{5}})}^{2} - 3^{2} = 0$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 5)}^{\frac{1}{5}}$ and $b = 3$ .

$({(x - 5)}^{\frac{1}{5}} + 3) ({(x - 5)}^{\frac{1}{5}} - 3) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x - 5)}^{\frac{1}{5}} + 3 = 0$

${(x - 5)}^{\frac{1}{5}} - 3 = 0$

Step 6

Set ${(x - 5)}^{\frac{1}{5}} + 3$ equal to $0$ and solve for $x$ .

Set ${(x - 5)}^{\frac{1}{5}} + 3$ equal to $0$ .

${(x - 5)}^{\frac{1}{5}} + 3 = 0$

Solve ${(x - 5)}^{\frac{1}{5}} + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

${(x - 5)}^{\frac{1}{5}} = - 3$

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${({(x - 5)}^{\frac{1}{5}})}^{5} = {(- 3)}^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x - 5)}^{\frac{1}{5}})}^{5}$ .

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

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

${(x - 5)}^{\frac{1}{5} \cdot 5} = {(- 3)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x - 5)}^{\frac{1}{5} \cdot 5} = {(- 3)}^{5}$

Rewrite the expression.

${(x - 5)}^{1} = {(- 3)}^{5}$

Simplify.

$x - 5 = {(- 3)}^{5}$

Simplify the right side.

Raise $- 3$ to the power of $5$ .

$x - 5 = - 243$

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

Add $5$ to both sides of the equation.

$x = - 243 + 5$

Add $- 243$ and $5$ .

$x = - 238$

Step 7

Set ${(x - 5)}^{\frac{1}{5}} - 3$ equal to $0$ and solve for $x$ .

Set ${(x - 5)}^{\frac{1}{5}} - 3$ equal to $0$ .

${(x - 5)}^{\frac{1}{5}} - 3 = 0$

Solve ${(x - 5)}^{\frac{1}{5}} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

${(x - 5)}^{\frac{1}{5}} = 3$

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${({(x - 5)}^{\frac{1}{5}})}^{5} = 3^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x - 5)}^{\frac{1}{5}})}^{5}$ .

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

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

${(x - 5)}^{\frac{1}{5} \cdot 5} = 3^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x - 5)}^{\frac{1}{5} \cdot 5} = 3^{5}$

Rewrite the expression.

${(x - 5)}^{1} = 3^{5}$

Simplify.

$x - 5 = 3^{5}$

Simplify the right side.

Raise $3$ to the power of $5$ .

$x - 5 = 243$

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

Add $5$ to both sides of the equation.

$x = 243 + 5$

Add $243$ and $5$ .

$x = 248$

Step 8

The final solution is all the values that make $({(x - 5)}^{\frac{1}{5}} + 3) ({(x - 5)}^{\frac{1}{5}} - 3) = 0$ true.

$x = - 238, 248$

Solve by Factoring (x-5)^(2/5)=9

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