Precalculus Examples

$x^{\frac{2}{3}} - \frac{16}{49} = 0$

Step 1

Add $\frac{16}{49}$ to both sides of the equation.

$x^{\frac{2}{3}} = \frac{16}{49}$

Step 2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(\frac{16}{49})}^{\frac{3}{2}}$

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

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Step 3.1.1.1

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

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Step 3.1.1.1.1

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

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{16}{49})}^{\frac{3}{2}}$

Step 3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{16}{49})}^{\frac{3}{2}}$

Step 3.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm {(\frac{16}{49})}^{\frac{3}{2}}$

Step 3.1.1.1.3

Cancel the common factor of $3$ .

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Step 3.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm {(\frac{16}{49})}^{\frac{3}{2}}$

Step 3.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm {(\frac{16}{49})}^{\frac{3}{2}}$

Step 3.1.1.2

Simplify.

$x = \pm {(\frac{16}{49})}^{\frac{3}{2}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\pm {(\frac{16}{49})}^{\frac{3}{2}}$ .

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Step 3.2.1.1

Apply the product rule to $\frac{16}{49}$ .

$x = \pm \frac{16^{\frac{3}{2}}}{49^{\frac{3}{2}}}$

Step 3.2.1.2

Simplify the numerator.

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Step 3.2.1.2.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \frac{{(4^{2})}^{\frac{3}{2}}}{49^{\frac{3}{2}}}$

Step 3.2.1.2.2

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

$x = \pm \frac{4^{2 (\frac{3}{2})}}{49^{\frac{3}{2}}}$

Step 3.2.1.2.3

Cancel the common factor of $2$ .

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Step 3.2.1.2.3.1

Cancel the common factor.

$x = \pm \frac{4^{2 (\frac{3}{2})}}{49^{\frac{3}{2}}}$

Step 3.2.1.2.3.2

Rewrite the expression.

$x = \pm \frac{4^{3}}{49^{\frac{3}{2}}}$

Step 3.2.1.2.4

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

$x = \pm \frac{64}{49^{\frac{3}{2}}}$

Step 3.2.1.3

Simplify the denominator.

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Step 3.2.1.3.1

Rewrite $49$ as $7^{2}$ .

$x = \pm \frac{64}{{(7^{2})}^{\frac{3}{2}}}$

Step 3.2.1.3.2

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

$x = \pm \frac{64}{7^{2 (\frac{3}{2})}}$

Step 3.2.1.3.3

Cancel the common factor of $2$ .

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Step 3.2.1.3.3.1

Cancel the common factor.

$x = \pm \frac{64}{7^{2 (\frac{3}{2})}}$

Step 3.2.1.3.3.2

Rewrite the expression.

$x = \pm \frac{64}{7^{3}}$

Step 3.2.1.3.4

Raise $7$ to the power of $3$ .

$x = \pm \frac{64}{343}$

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.

$x = \frac{64}{343}$

Step 4.2

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

$x = - \frac{64}{343}$

Step 4.3

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

$x = \frac{64}{343}, - \frac{64}{343}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{64}{343}, - \frac{64}{343}$

Decimal Form:

$x = 0.18658892 \dots, - 0.18658892 \dots$