Algebra Examples

$3 x^{- \frac{3}{2}} + 40 = 121$

Step 1

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

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Step 1.1

Subtract $40$ from both sides of the equation.

$3 x^{- \frac{3}{2}} = 121 - 40$

Step 1.2

Subtract $40$ from $121$ .

$3 x^{- \frac{3}{2}} = 81$

Step 2

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

${(3 x^{- \frac{3}{2}})}^{- \frac{2}{3}} = 81^{- \frac{2}{3}}$

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

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Step 3.1.1.1

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

${(3 \frac{1}{x^{\frac{3}{2}}})}^{- \frac{2}{3}} = 81^{- \frac{2}{3}}$

Step 3.1.1.2

Combine $3$ and $\frac{1}{x^{\frac{3}{2}}}$ .

${(\frac{3}{x^{\frac{3}{2}}})}^{- \frac{2}{3}} = 81^{- \frac{2}{3}}$

Step 3.1.1.3

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

${(\frac{x^{\frac{3}{2}}}{3})}^{\frac{2}{3}} = 81^{- \frac{2}{3}}$

Step 3.1.1.4

Apply the product rule to $\frac{x^{\frac{3}{2}}}{3}$ .

$\frac{{(x^{\frac{3}{2}})}^{\frac{2}{3}}}{3^{\frac{2}{3}}} = 81^{- \frac{2}{3}}$

Step 3.1.1.5

Simplify the numerator.

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Step 3.1.1.5.1

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

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Step 3.1.1.5.1.1

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

$\frac{x^{\frac{3}{2} \cdot \frac{2}{3}}}{3^{\frac{2}{3}}} = 81^{- \frac{2}{3}}$

Step 3.1.1.5.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.5.1.2.1

Cancel the common factor.

$\frac{x^{\frac{3}{2} \cdot \frac{2}{3}}}{3^{\frac{2}{3}}} = 81^{- \frac{2}{3}}$

Step 3.1.1.5.1.2.2

Rewrite the expression.

$\frac{x^{\frac{1}{2} \cdot 2}}{3^{\frac{2}{3}}} = 81^{- \frac{2}{3}}$

Step 3.1.1.5.1.3

Cancel the common factor of $2$ .

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Step 3.1.1.5.1.3.1

Cancel the common factor.

$\frac{x^{\frac{1}{2} \cdot 2}}{3^{\frac{2}{3}}} = 81^{- \frac{2}{3}}$

Step 3.1.1.5.1.3.2

Rewrite the expression.

$\frac{x^{1}}{3^{\frac{2}{3}}} = 81^{- \frac{2}{3}}$

Step 3.1.1.5.2

Simplify.

$\frac{x}{3^{\frac{2}{3}}} = 81^{- \frac{2}{3}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

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

$\frac{x}{3^{\frac{2}{3}}} = \frac{1}{81^{\frac{2}{3}}}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $3^{\frac{2}{3}}$ .

$3^{\frac{2}{3}} \frac{x}{3^{\frac{2}{3}}} = 3^{\frac{2}{3}} \frac{1}{81^{\frac{2}{3}}}$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $3^{\frac{2}{3}}$ .

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Step 4.2.1.1.1

Cancel the common factor.

$3^{\frac{2}{3}} \frac{x}{3^{\frac{2}{3}}} = 3^{\frac{2}{3}} \frac{1}{81^{\frac{2}{3}}}$

Step 4.2.1.1.2

Rewrite the expression.

$x = 3^{\frac{2}{3}} \frac{1}{81^{\frac{2}{3}}}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Combine $3^{\frac{2}{3}}$ and $\frac{1}{81^{\frac{2}{3}}}$ .

$x = \frac{3^{\frac{2}{3}}}{81^{\frac{2}{3}}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3^{\frac{2}{3}}}{81^{\frac{2}{3}}}$

Decimal Form:

$x = 0.\overline{1}$