Solve for n (n-27)^(3/2)=64

${(n - 27)}^{\frac{3}{2}} = 64$

Step 1

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${({(n - 27)}^{\frac{3}{2}})}^{\frac{2}{3}} = 64^{\frac{2}{3}}$

Step 2

Simplify the exponent.

Simplify the left side.

Simplify ${({(n - 27)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Multiply the exponents in ${({(n - 27)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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

${(n - 27)}^{\frac{3}{2} \cdot \frac{2}{3}} = 64^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(n - 27)}^{\frac{3}{2} \cdot \frac{2}{3}} = 64^{\frac{2}{3}}$

Rewrite the expression.

${(n - 27)}^{\frac{1}{2} \cdot 2} = 64^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(n - 27)}^{\frac{1}{2} \cdot 2} = 64^{\frac{2}{3}}$

Rewrite the expression.

${(n - 27)}^{1} = 64^{\frac{2}{3}}$

Simplify.

$n - 27 = 64^{\frac{2}{3}}$

Simplify the right side.

Simplify $64^{\frac{2}{3}}$ .

Simplify the expression.

Rewrite $64$ as $4^{3}$ .

$n - 27 = {(4^{3})}^{\frac{2}{3}}$

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

$n - 27 = 4^{3 (\frac{2}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$n - 27 = 4^{3 (\frac{2}{3})}$

Rewrite the expression.

$n - 27 = 4^{2}$

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

$n - 27 = 16$

Step 3

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

Add $27$ to both sides of the equation.

$n = 16 + 27$

Add $16$ and $27$ .

$n = 43$