Solve for x 9 cube root of (x-7)^4=9

$9 \sqrt[3]{{(x - 7)}^{4}} = 9$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${(9 \sqrt[3]{{(x - 7)}^{4}})}^{3} = 9^{3}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(x - 7)}^{4}}$ as ${(x - 7)}^{\frac{4}{3}}$ .

${(9 {(x - 7)}^{\frac{4}{3}})}^{3} = 9^{3}$

Simplify the left side.

Simplify ${(9 {(x - 7)}^{\frac{4}{3}})}^{3}$ .

Apply the product rule to $9 {(x - 7)}^{\frac{4}{3}}$ .

$9^{3} {({(x - 7)}^{\frac{4}{3}})}^{3} = 9^{3}$

Raise $9$ to the power of $3$ .

$729 {({(x - 7)}^{\frac{4}{3}})}^{3} = 9^{3}$

Multiply the exponents in ${({(x - 7)}^{\frac{4}{3}})}^{3}$ .

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

$729 {(x - 7)}^{\frac{4}{3} \cdot 3} = 9^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$729 {(x - 7)}^{\frac{4}{3} \cdot 3} = 9^{3}$

Rewrite the expression.

$729 {(x - 7)}^{4} = 9^{3}$

Simplify the right side.

Raise $9$ to the power of $3$ .

$729 {(x - 7)}^{4} = 729$

Step 3

Solve for $x$ .

Divide each term in $729 {(x - 7)}^{4} = 729$ by $729$ and simplify.

Divide each term in $729 {(x - 7)}^{4} = 729$ by $729$ .

$\frac{729 {(x - 7)}^{4}}{729} = \frac{729}{729}$

Simplify the left side.

Cancel the common factor of $729$ .

Cancel the common factor.

$\frac{729 {(x - 7)}^{4}}{729} = \frac{729}{729}$

Divide ${(x - 7)}^{4}$ by $1$ .

${(x - 7)}^{4} = \frac{729}{729}$

Simplify the right side.

Divide $729$ by $729$ .

${(x - 7)}^{4} = 1$

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

$x - 7 = \pm \sqrt[4]{1}$

Any root of $1$ is $1$ .

$x - 7 = \pm 1$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x - 7 = 1$

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

Add $7$ to both sides of the equation.

$x = 1 + 7$

Add $1$ and $7$ .

$x = 8$

Next, use the negative value of the $\pm$ to find the second solution.

$x - 7 = - 1$

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

Add $7$ to both sides of the equation.

$x = - 1 + 7$

Add $- 1$ and $7$ .

$x = 6$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 8, 6$

Solve for x 9 cube root of (x-7)^4=9

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