Algebra Examples

$- 7 + {(x^{2} - 19)}^{\frac{3}{4}} = 20$

Step 1

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

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Step 1.1

Add $7$ to both sides of the equation.

${(x^{2} - 19)}^{\frac{3}{4}} = 20 + 7$

Step 1.2

Add $20$ and $7$ .

${(x^{2} - 19)}^{\frac{3}{4}} = 27$

Step 2

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

${({(x^{2} - 19)}^{\frac{3}{4}})}^{\frac{4}{3}} = 27^{\frac{4}{3}}$

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 ${({(x^{2} - 19)}^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 3.1.1.1

Multiply the exponents in ${({(x^{2} - 19)}^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 3.1.1.1.1

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

${(x^{2} - 19)}^{\frac{3}{4} \cdot \frac{4}{3}} = 27^{\frac{4}{3}}$

Step 3.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.1.2.1

Cancel the common factor.

${(x^{2} - 19)}^{\frac{3}{4} \cdot \frac{4}{3}} = 27^{\frac{4}{3}}$

Step 3.1.1.1.2.2

Rewrite the expression.

${(x^{2} - 19)}^{\frac{1}{4} \cdot 4} = 27^{\frac{4}{3}}$

Step 3.1.1.1.3

Cancel the common factor of $4$ .

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Step 3.1.1.1.3.1

Cancel the common factor.

${(x^{2} - 19)}^{\frac{1}{4} \cdot 4} = 27^{\frac{4}{3}}$

Step 3.1.1.1.3.2

Rewrite the expression.

${(x^{2} - 19)}^{1} = 27^{\frac{4}{3}}$

Step 3.1.1.2

Simplify.

$x^{2} - 19 = 27^{\frac{4}{3}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $27^{\frac{4}{3}}$ .

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Step 3.2.1.1

Simplify the expression.

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Step 3.2.1.1.1

Rewrite $27$ as $3^{3}$ .

$x^{2} - 19 = {(3^{3})}^{\frac{4}{3}}$

Step 3.2.1.1.2

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

$x^{2} - 19 = 3^{3 (\frac{4}{3})}$

Step 3.2.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.2.1

Cancel the common factor.

$x^{2} - 19 = 3^{3 (\frac{4}{3})}$

Step 3.2.1.2.2

Rewrite the expression.

$x^{2} - 19 = 3^{4}$

Step 3.2.1.3

Raise $3$ to the power of $4$ .

$x^{2} - 19 = 81$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $19$ to both sides of the equation.

$x^{2} = 81 + 19$

Step 4.1.2

Add $81$ and $19$ .

$x^{2} = 100$

Step 4.2

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

$x = \pm \sqrt{100}$

Step 4.3

Simplify $\pm \sqrt{100}$ .

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Step 4.3.1

Rewrite $100$ as $10^{2}$ .

$x = \pm \sqrt{10^{2}}$

Step 4.3.2

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

$x = \pm 10$

Step 4.4

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

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Step 4.4.1

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

$x = 10$

Step 4.4.2

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

$x = - 10$

Step 4.4.3

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

$x = 10, - 10$