Algebra Examples

$4^{x + 3} = \sqrt{8^{x + 5}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{8^{x + 5}} = 4^{x + 3}$

Step 2

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

${\sqrt{8^{x + 5}}}^{2} = {(4^{x + 3})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8^{x + 5}}$ as $8^{\frac{x + 5}{2}}$ .

${(8^{\frac{x + 5}{2}})}^{2} = {(4^{x + 3})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply the exponents in ${(8^{\frac{x + 5}{2}})}^{2}$ .

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Step 3.2.1.1

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

$8^{\frac{x + 5}{2} \cdot 2} = {(4^{x + 3})}^{2}$

Step 3.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.2.1

Cancel the common factor.

$8^{\frac{x + 5}{2} \cdot 2} = {(4^{x + 3})}^{2}$

Step 3.2.1.2.2

Rewrite the expression.

$8^{x + 5} = {(4^{x + 3})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the exponents in ${(4^{x + 3})}^{2}$ .

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Step 3.3.1.1

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

$8^{x + 5} = 4^{(x + 3) \cdot 2}$

Step 3.3.1.2

Apply the distributive property.

$8^{x + 5} = 4^{x \cdot 2 + 3 \cdot 2}$

Step 3.3.1.3

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

$8^{x + 5} = 4^{2 \cdot x + 3 \cdot 2}$

Step 3.3.1.4

Multiply $3$ by $2$ .

$8^{x + 5} = 4^{2 x + 6}$

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Solve for $x$ .

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Step 4.3.1

Simplify $3 (x + 5)$ .

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Step 4.3.1.1

Rewrite.

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

Step 4.3.1.2

Simplify by adding zeros.

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

Step 4.3.1.3

Apply the distributive property.

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

Step 4.3.1.4

Multiply $3$ by $5$ .

$3 x + 15 = 2 (2 x + 6)$

Step 4.3.2

Simplify $2 (2 x + 6)$ .

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Step 4.3.2.1

Apply the distributive property.

$3 x + 15 = 2 (2 x) + 2 \cdot 6$

Step 4.3.2.2

Multiply.

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Step 4.3.2.2.1

Multiply $2$ by $2$ .

$3 x + 15 = 4 x + 2 \cdot 6$

Step 4.3.2.2.2

Multiply $2$ by $6$ .

$3 x + 15 = 4 x + 12$

Step 4.3.3

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

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Step 4.3.3.1

Subtract $4 x$ from both sides of the equation.

$3 x + 15 - 4 x = 12$

Step 4.3.3.2

Subtract $4 x$ from $3 x$ .

$- x + 15 = 12$

Step 4.3.4

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

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Step 4.3.4.1

Subtract $15$ from both sides of the equation.

$- x = 12 - 15$

Step 4.3.4.2

Subtract $15$ from $12$ .

$- x = - 3$

Step 4.3.5

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 4.3.5.1

Divide each term in $- x = - 3$ by $- 1$ .

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

Step 4.3.5.2

Simplify the left side.

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Step 4.3.5.2.1

Dividing two negative values results in a positive value.

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

Step 4.3.5.2.2

Divide $x$ by $1$ .

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

Step 4.3.5.3

Simplify the right side.

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Step 4.3.5.3.1

Divide $- 3$ by $- 1$ .

$x = 3$