Algebra Examples

$\sqrt{4^{5 (x - 2)}} = 16^{3 - x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4^{5 (x - 2)}}$ as $4^{\frac{5 (x - 2)}{2}}$ .

$4^{\frac{5 (x - 2)}{2}} = 16^{3 - x}$

Step 2

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

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

Step 3

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

$\frac{5 (x - 2)}{2} = 2 (3 - x)$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $2$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $\frac{5 (x - 2)}{2} \cdot 2$ .

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Step 4.2.1.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1.1

Cancel the common factor.

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

Step 4.2.1.1.1.2

Rewrite the expression.

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

Step 4.2.1.1.2

Apply the distributive property.

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

Step 4.2.1.1.3

Multiply $5$ by $- 2$ .

$5 x - 10 = 2 (3 - x) \cdot 2$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $2 (3 - x) \cdot 2$ .

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Step 4.2.2.1.1

Apply the distributive property.

$5 x - 10 = (2 \cdot 3 + 2 (- x)) \cdot 2$

Step 4.2.2.1.2

Multiply.

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Step 4.2.2.1.2.1

Multiply $2$ by $3$ .

$5 x - 10 = (6 + 2 (- x)) \cdot 2$

Step 4.2.2.1.2.2

Multiply $- 1$ by $2$ .

$5 x - 10 = (6 - 2 x) \cdot 2$

Step 4.2.2.1.3

Apply the distributive property.

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

Step 4.2.2.1.4

Simplify the expression.

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Step 4.2.2.1.4.1

Multiply $6$ by $2$ .

$5 x - 10 = 12 - 2 x \cdot 2$

Step 4.2.2.1.4.2

Multiply $2$ by $- 2$ .

$5 x - 10 = 12 - 4 x$

Step 4.2.2.1.4.3

Reorder $12$ and $- 4 x$ .

$5 x - 10 = - 4 x + 12$

Step 4.3

Solve for $x$ .

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Step 4.3.1

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

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Step 4.3.1.1

Add $4 x$ to both sides of the equation.

$5 x - 10 + 4 x = 12$

Step 4.3.1.2

Add $5 x$ and $4 x$ .

$9 x - 10 = 12$

Step 4.3.2

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

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Step 4.3.2.1

Add $10$ to both sides of the equation.

$9 x = 12 + 10$

Step 4.3.2.2

Add $12$ and $10$ .

$9 x = 22$

Step 4.3.3

Divide each term in $9 x = 22$ by $9$ and simplify.

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Step 4.3.3.1

Divide each term in $9 x = 22$ by $9$ .

$\frac{9 x}{9} = \frac{22}{9}$

Step 4.3.3.2

Simplify the left side.

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Step 4.3.3.2.1

Cancel the common factor of $9$ .

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Step 4.3.3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{22}{9}$

Step 4.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{22}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{22}{9}$

Decimal Form:

$x = 2.\overline{4}$

Mixed Number Form:

$x = 2 \frac{4}{9}$