Algebra Examples

${(2 x)}^{- 2} = 16$

Step 1

Simplify ${(2 x)}^{- 2}$ .

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

Apply the product rule to $2 x$ .

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

Step 1.2.2

Raise $2$ to the power of $2$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$4 x^{2}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$4 x^{2}$

Step 3

Multiply each term in $\frac{1}{4 x^{2}} = 16$ by $4 x^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{4 x^{2}} = 16$ by $4 x^{2}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

Cancel the common factor of $4$ .

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Step 3.2.2.1

Factor $4$ out of $4 x^{2}$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Cancel the common factor of $x^{2}$ .

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Step 3.2.3.1

Cancel the common factor.

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

Step 3.2.3.2

Rewrite the expression.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $4$ by $16$ .

$1 = 64 x^{2}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $64 x^{2} = 1$ .

$64 x^{2} = 1$

Step 4.2

Divide each term in $64 x^{2} = 1$ by $64$ and simplify.

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Step 4.2.1

Divide each term in $64 x^{2} = 1$ by $64$ .

$\frac{64 x^{2}}{64} = \frac{1}{64}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $64$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{64 x^{2}}{64} = \frac{1}{64}$

Step 4.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{64}$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{64}}$

Step 4.4

Simplify $\pm \sqrt{\frac{1}{64}}$ .

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Step 4.4.1

Rewrite $\sqrt{\frac{1}{64}}$ as $\frac{\sqrt{1}}{\sqrt{64}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{64}}$

Step 4.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{64}}$

Step 4.4.3

Simplify the denominator.

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Step 4.4.3.1

Rewrite $64$ as $8^{2}$ .

$x = \pm \frac{1}{\sqrt{8^{2}}}$

Step 4.4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{1}{8}$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.125, - 0.125$