Solve by Factoring (x-3)^(5/2)=243

${(x - 3)}^{\frac{5}{2}} = 243$

Step 1

Subtract $243$ from both sides of the equation.

${(x - 3)}^{\frac{5}{2}} - 243 = 0$

Step 2

Add $243$ to both sides of the equation.

${(x - 3)}^{\frac{5}{2}} = 243$

Step 3

Raise each side of the equation to the power of $\frac{2}{5}$ to eliminate the fractional exponent on the left side.

${({(x - 3)}^{\frac{5}{2}})}^{\frac{2}{5}} = 243^{\frac{2}{5}}$

Step 4

Simplify the exponent.

Simplify the left side.

Simplify ${({(x - 3)}^{\frac{5}{2}})}^{\frac{2}{5}}$ .

Multiply the exponents in ${({(x - 3)}^{\frac{5}{2}})}^{\frac{2}{5}}$ .

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

${(x - 3)}^{\frac{5}{2} \cdot \frac{2}{5}} = 243^{\frac{2}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x - 3)}^{\frac{5}{2} \cdot \frac{2}{5}} = 243^{\frac{2}{5}}$

Rewrite the expression.

${(x - 3)}^{\frac{1}{2} \cdot 2} = 243^{\frac{2}{5}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 3)}^{\frac{1}{2} \cdot 2} = 243^{\frac{2}{5}}$

Rewrite the expression.

${(x - 3)}^{1} = 243^{\frac{2}{5}}$

Simplify.

$x - 3 = 243^{\frac{2}{5}}$

Simplify the right side.

Simplify $243^{\frac{2}{5}}$ .

Simplify the expression.

Rewrite $243$ as $3^{5}$ .

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

$x - 3 = 3^{2}$

Raise $3$ to the power of $2$ .

$x - 3 = 9$

Step 5

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

Add $3$ to both sides of the equation.

$x = 9 + 3$

Add $9$ and $3$ .

$x = 12$

Solve by Factoring (x-3)^(5/2)=243

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