Solve for x 16(1/2)^(x/3)=4

$16 {(\frac{1}{2})}^{\frac{x}{3}} = 4$

Step 1

Apply the product rule to $\frac{1}{2}$ .

$16 \frac{1^{\frac{x}{3}}}{2^{\frac{x}{3}}} = 4$

Step 2

One to any power is one.

$16 \frac{1}{2^{\frac{x}{3}}} = 4$

Step 3

Move $2^{\frac{x}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$16 \cdot 2^{- \frac{x}{3}} = 4$

Step 4

Rewrite $16$ as $2^{4}$ .

$2^{4} \cdot 2^{- \frac{x}{3}} = 4$

Step 5

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

$2^{4 - \frac{x}{3}} = 4$

Step 6

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

$2^{4 - \frac{x}{3}} = 2^{2}$

Step 7

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

$4 - \frac{x}{3} = 2$

Step 8

Solve for $x$ .

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

Subtract $4$ from both sides of the equation.

$- \frac{x}{3} = 2 - 4$

Subtract $4$ from $2$ .

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

Multiply both sides of the equation by $- 3$ .

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

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{x}{3}$ into the numerator.

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

$3 \cdot - 1 \frac{- x}{3} = - 3 \cdot - 2$

Rewrite the expression.

$- - x = - 3 \cdot - 2$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 x = - 3 \cdot - 2$

Multiply $x$ by $1$ .

$x = - 3 \cdot - 2$

Simplify the right side.

Multiply $- 3$ by $- 2$ .

$x = 6$

Solve for x 16(1/2)^(x/3)=4

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