Solve for x 7+x^(3/2)=71

$7 + x^{\frac{3}{2}} = 71$

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

Subtract $7$ from both sides of the equation.

$x^{\frac{3}{2}} - 71 = - 7$

Add $71$ to both sides of the equation.

$x^{\frac{3}{2}} = - 7 + 71$

Add $- 7$ and $71$ .

$x^{\frac{3}{2}} = 64$

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

${(x^{\frac{3}{2}})}^{\frac{2}{3}} = 64^{\frac{2}{3}}$

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{3}{2} \cdot \frac{2}{3}} = 64^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{3}{2} \cdot \frac{2}{3}} = 64^{\frac{2}{3}}$

Rewrite the expression.

$x^{\frac{1}{2} \cdot 2} = 64^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 64^{\frac{2}{3}}$

Rewrite the expression.

$x^{1} = 64^{\frac{2}{3}}$

Simplify.

$x = 64^{\frac{2}{3}}$

Simplify the right side.

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

Simplify the expression.

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

$x = {(4^{3})}^{\frac{2}{3}}$

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

$x = 4^{3 (\frac{2}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x = 4^{3 (\frac{2}{3})}$

Rewrite the expression.

$x = 4^{2}$

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

$x = 16$