Solve for x square root of (8^(x-1))/(2^x)=32^(x+3)

$\sqrt{\frac{8^{x - 1}}{2^{x}}} = 32^{x + 3}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{8^{x - 1}}{2^{x}}}}^{2} = {(32^{x + 3})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{8^{x - 1}}{2^{x}}}$ as ${(\frac{8^{x - 1}}{2^{x}})}^{\frac{1}{2}}$ .

${({(\frac{8^{x - 1}}{2^{x}})}^{\frac{1}{2}})}^{2} = {(32^{x + 3})}^{2}$

Simplify the left side.

Simplify ${({(\frac{8^{x - 1}}{2^{x}})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{8^{x - 1}}{2^{x}})}^{\frac{1}{2}})}^{2}$ .

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

${(\frac{8^{x - 1}}{2^{x}})}^{\frac{1}{2} \cdot 2} = {(32^{x + 3})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{8^{x - 1}}{2^{x}})}^{\frac{1}{2} \cdot 2} = {(32^{x + 3})}^{2}$

Rewrite the expression.

${(\frac{8^{x - 1}}{2^{x}})}^{1} = {(32^{x + 3})}^{2}$

Simplify.

$\frac{8^{x - 1}}{2^{x}} = {(32^{x + 3})}^{2}$

Simplify the right side.

Multiply the exponents in ${(32^{x + 3})}^{2}$ .

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

$\frac{8^{x - 1}}{2^{x}} = 32^{(x + 3) \cdot 2}$

Apply the distributive property.

$\frac{8^{x - 1}}{2^{x}} = 32^{x \cdot 2 + 3 \cdot 2}$

Move $2$ to the left of $x$ .

$\frac{8^{x - 1}}{2^{x}} = 32^{2 \cdot x + 3 \cdot 2}$

Multiply $3$ by $2$ .

$\frac{8^{x - 1}}{2^{x}} = 32^{2 x + 6}$

Step 3

Solve for $x$ .

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

$8^{x - 1} \cdot 2^{- x} = 32^{2 x + 6}$

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

${(2^{3})}^{x - 1} \cdot 2^{- x} = 32^{2 x + 6}$

Multiply the exponents in ${(2^{3})}^{x - 1}$ .

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

$2^{3 (x - 1)} \cdot 2^{- x} = 32^{2 x + 6}$

Apply the distributive property.

$2^{3 x + 3 \cdot - 1} \cdot 2^{- x} = 32^{2 x + 6}$

Multiply $3$ by $- 1$ .

$2^{3 x - 3} \cdot 2^{- x} = 32^{2 x + 6}$

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

$2^{3 x - 3 - x} = 32^{2 x + 6}$

Subtract $x$ from $3 x$ .

$2^{2 x - 3} = 32^{2 x + 6}$

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

$2^{2 x - 3} = 2^{5 (2 x + 6)}$

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

$2 x - 3 = 5 (2 x + 6)$

Solve for $x$ .

Simplify $5 (2 x + 6)$ .

Rewrite.

$2 x - 3 = 0 + 0 + 5 (2 x + 6)$

Simplify by adding zeros.

$2 x - 3 = 5 (2 x + 6)$

Apply the distributive property.

$2 x - 3 = 5 (2 x) + 5 \cdot 6$

Multiply.

Multiply $2$ by $5$ .

$2 x - 3 = 10 x + 5 \cdot 6$

Multiply $5$ by $6$ .

$2 x - 3 = 10 x + 30$

Move all terms containing $x$ to the left side of the equation.

Subtract $10 x$ from both sides of the equation.

$2 x - 3 - 10 x = 30$

Subtract $10 x$ from $2 x$ .

$- 8 x - 3 = 30$

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

Add $3$ to both sides of the equation.

$- 8 x = 30 + 3$

Add $30$ and $3$ .

$- 8 x = 33$

Divide each term in $- 8 x = 33$ by $- 8$ and simplify.

Divide each term in $- 8 x = 33$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{33}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{33}{- 8}$

Divide $x$ by $1$ .

$x = \frac{33}{- 8}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{33}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{33}{8}$

Decimal Form:

$x = - 4.125$

Mixed Number Form:

$x = - 4 \frac{1}{8}$

Solve for x square root of (8^(x-1))/(2^x)=32^(x+3)

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