Trigonometry Examples

2 x^{- \frac{2}{5}} - 4 = 4

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

Step 1

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 1.1

Add

4

$4$ to both sides of the equation.

2 x^{- \frac{2}{5}} = 4 + 4

$2 x^{- \frac{2}{5}} = 4 + 4$

Step 1.2

Add

4

$4$ and

4

$4$ .

2 x^{- \frac{2}{5}} = 8

$2 x^{- \frac{2}{5}} = 8$

2 x^{- \frac{2}{5}} = 8

$2 x^{- \frac{2}{5}} = 8$

Step 2

Raise each side of the equation to the power of

- \frac{5}{2}

$- \frac{5}{2}$ to eliminate the fractional exponent on the left side.

{(2 x^{- \frac{2}{5}})}^{- \frac{5}{2}} = \pm 8^{- \frac{5}{2}}

${(2 x^{- \frac{2}{5}})}^{- \frac{5}{2}} = \pm 8^{- \frac{5}{2}}$

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify

{(2 x^{- \frac{2}{5}})}^{- \frac{5}{2}}

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

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Step 3.1.1.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

{(2 \frac{1}{x^{\frac{2}{5}}})}^{- \frac{5}{2}} = \pm 8^{- \frac{5}{2}}

${(2 \frac{1}{x^{\frac{2}{5}}})}^{- \frac{5}{2}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.2

Combine

2

$2$ and

\frac{1}{x^{\frac{2}{5}}}

$\frac{1}{x^{\frac{2}{5}}}$ .

{(\frac{2}{x^{\frac{2}{5}}})}^{- \frac{5}{2}} = \pm 8^{- \frac{5}{2}}

${(\frac{2}{x^{\frac{2}{5}}})}^{- \frac{5}{2}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.3

Change the sign of the exponent by rewriting the base as its reciprocal.

{(\frac{x^{\frac{2}{5}}}{2})}^{\frac{5}{2}} = \pm 8^{- \frac{5}{2}}

${(\frac{x^{\frac{2}{5}}}{2})}^{\frac{5}{2}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.4

Apply the product rule to

\frac{x^{\frac{2}{5}}}{2}

$\frac{x^{\frac{2}{5}}}{2}$ .

\frac{{(x^{\frac{2}{5}})}^{\frac{5}{2}}}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}

$\frac{{(x^{\frac{2}{5}})}^{\frac{5}{2}}}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.5

Simplify the numerator.

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Step 3.1.1.5.1

Multiply the exponents in

{(x^{\frac{2}{5}})}^{\frac{5}{2}}

${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 3.1.1.5.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{x^{\frac{2}{5} \cdot \frac{5}{2}}}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}

$\frac{x^{\frac{2}{5} \cdot \frac{5}{2}}}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.5.1.2

Cancel the common factor of

2

$2$ .

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Step 3.1.1.5.1.2.1

Cancel the common factor.

$\frac{x^{\frac{2}{5} \cdot \frac{5}{2}}}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.5.1.2.2

Rewrite the expression.

$\frac{x^{\frac{1}{5} \cdot 5}}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.5.1.3

Cancel the common factor of

$5$ .

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Step 3.1.1.5.1.3.1

Cancel the common factor.

$\frac{x^{\frac{1}{5} \cdot 5}}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.5.1.3.2

Rewrite the expression.

$\frac{x^{1}}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}$

Step 3.1.1.5.2

Simplify.

$\frac{x}{2^{\frac{5}{2}}} = \pm 8^{- \frac{5}{2}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x}{2^{\frac{5}{2}}} = \pm \frac{1}{8^{\frac{5}{2}}}$

Step 4

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

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Step 4.1

First, use the positive value of the

$\pm$ to find the first solution.

$\frac{x}{2^{\frac{5}{2}}} = \frac{1}{8^{\frac{5}{2}}}$

Step 4.2

Multiply both sides of the equation by

$2^{\frac{5}{2}}$ .

$2^{\frac{5}{2}} \frac{x}{2^{\frac{5}{2}}} = 2^{\frac{5}{2}} \frac{1}{8^{\frac{5}{2}}}$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Cancel the common factor of

$2^{\frac{5}{2}}$ .

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Step 4.3.1.1.1

Cancel the common factor.

$2^{\frac{5}{2}} \frac{x}{2^{\frac{5}{2}}} = 2^{\frac{5}{2}} \frac{1}{8^{\frac{5}{2}}}$

Step 4.3.1.1.2

Rewrite the expression.

$x = 2^{\frac{5}{2}} \frac{1}{8^{\frac{5}{2}}}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Combine

$2^{\frac{5}{2}}$ and

$\frac{1}{8^{\frac{5}{2}}}$ .

$x = \frac{2^{\frac{5}{2}}}{8^{\frac{5}{2}}}$

Step 4.4

Next, use the negative value of the

$\pm$ to find the second solution.

$\frac{x}{2^{\frac{5}{2}}} = - \frac{1}{8^{\frac{5}{2}}}$

Step 4.5

Multiply both sides of the equation by

$2^{\frac{5}{2}}$ .

$2^{\frac{5}{2}} \frac{x}{2^{\frac{5}{2}}} = 2^{\frac{5}{2}} (- \frac{1}{8^{\frac{5}{2}}})$

Step 4.6

Simplify both sides of the equation.

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Step 4.6.1

Simplify the left side.

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Step 4.6.1.1

Cancel the common factor of

$2^{\frac{5}{2}}$ .

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Step 4.6.1.1.1

Cancel the common factor.

$2^{\frac{5}{2}} \frac{x}{2^{\frac{5}{2}}} = 2^{\frac{5}{2}} (- \frac{1}{8^{\frac{5}{2}}})$

Step 4.6.1.1.2

Rewrite the expression.

$x = 2^{\frac{5}{2}} (- \frac{1}{8^{\frac{5}{2}}})$

Step 4.6.2

Simplify the right side.

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Step 4.6.2.1

Combine

$2^{\frac{5}{2}}$ and

$\frac{1}{8^{\frac{5}{2}}}$ .

$x = - \frac{2^{\frac{5}{2}}}{8^{\frac{5}{2}}}$

Step 4.7

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

$x = \frac{2^{\frac{5}{2}}}{8^{\frac{5}{2}}}, - \frac{2^{\frac{5}{2}}}{8^{\frac{5}{2}}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2^{\frac{5}{2}}}{8^{\frac{5}{2}}}, - \frac{2^{\frac{5}{2}}}{8^{\frac{5}{2}}}$

Decimal Form:

$x = 0.03125, - 0.03125$