Solve for x 2=(32^(1/3))(2^(x/3))

$2 = (32^{\frac{1}{3}}) (2^{\frac{x}{3}})$

Step 1

Rewrite the equation as $32^{\frac{1}{3}} \cdot 2^{\frac{x}{3}} = 2$ .

$32^{\frac{1}{3}} \cdot 2^{\frac{x}{3}} = 2$

Step 2

Rewrite $32$ as $2^{5}$ .

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

Step 3

Multiply the exponents in ${(2^{5})}^{\frac{1}{3}}$ .

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

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

Combine $5$ and $\frac{1}{3}$ .

$2^{\frac{5}{3}} \cdot 2^{\frac{x}{3}} = 2$

Step 4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{\frac{5}{3} + \frac{x}{3}} = 2$

Step 5

Create equivalent expressions in the equation that all have equal bases.

$2^{\frac{5}{3} + \frac{x}{3}} = 2^{1}$

Step 6

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$\frac{5}{3} + \frac{x}{3} = 1$

Step 7

Solve for $x$ .

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

Subtract $\frac{5}{3}$ from both sides of the equation.

$\frac{x}{3} = 1 - \frac{5}{3}$

Write $1$ as a fraction with a common denominator.

$\frac{x}{3} = \frac{3}{3} - \frac{5}{3}$

Combine the numerators over the common denominator.

$\frac{x}{3} = \frac{3 - 5}{3}$

Subtract $5$ from $3$ .

$\frac{x}{3} = \frac{- 2}{3}$

Move the negative in front of the fraction.

$\frac{x}{3} = - \frac{2}{3}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - 2$