Solve for a 2^5*8^(2/3)=(1/2)^(3-2a)

$2^{5} \cdot 8^{\frac{2}{3}} = {(\frac{1}{2})}^{3 - 2 a}$

Step 1

Rewrite the equation as ${(\frac{1}{2})}^{3 - 2 a} = 2^{5} \cdot 8^{\frac{2}{3}}$ .

${(\frac{1}{2})}^{3 - 2 a} = 2^{5} \cdot 8^{\frac{2}{3}}$

Step 2

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

$\frac{1^{3 - 2 a}}{2^{3 - 2 a}} = 2^{5} \cdot 8^{\frac{2}{3}}$

Step 3

One to any power is one.

$\frac{1}{2^{3 - 2 a}} = 2^{5} \cdot 8^{\frac{2}{3}}$

Step 4

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

$2^{- (3 - 2 a)} = 2^{5} \cdot 8^{\frac{2}{3}}$

Step 5

Rewrite $8$ as $2^{3}$ .

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

Step 6

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$2^{- (3 - 2 a)} = 2^{5} \cdot 2^{2}$

Step 7

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

$2^{- (3 - 2 a)} = 2^{5 + 2}$

Step 8

Add $5$ and $2$ .

$2^{- (3 - 2 a)} = 2^{7}$

Step 9

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

$- (3 - 2 a) = 7$

Step 10

Solve for $a$ .

Simplify $- (3 - 2 a)$ .

Apply the distributive property.

$- 1 \cdot 3 - (- 2 a) = 7$

Multiply.

Multiply $- 1$ by $3$ .

$- 3 - (- 2 a) = 7$

Multiply $- 2$ by $- 1$ .

$- 3 + 2 a = 7$

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

Add $3$ to both sides of the equation.

$2 a = 7 + 3$

Add $7$ and $3$ .

$2 a = 10$

Divide each term in $2 a = 10$ by $2$ and simplify.

Divide each term in $2 a = 10$ by $2$ .

$\frac{2 a}{2} = \frac{10}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{10}{2}$

Divide $a$ by $1$ .

$a = \frac{10}{2}$

Simplify the right side.

Divide $10$ by $2$ .

$a = 5$

Solve for a 2^5*8^(2/3)=(1/2)^(3-2a)

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