Algebra Examples

$- 10 + 3 {(4 x)}^{- \frac{1}{6}} = - \frac{17}{2}$

Step 1

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

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Step 1.1

Add $10$ to both sides of the equation.

$3 {(4 x)}^{- \frac{1}{6}} = - \frac{17}{2} + 10$

Step 1.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 {(4 x)}^{- \frac{1}{6}} = - \frac{17}{2} + 10 \cdot \frac{2}{2}$

Step 1.3

Combine $10$ and $\frac{2}{2}$ .

$3 {(4 x)}^{- \frac{1}{6}} = - \frac{17}{2} + \frac{10 \cdot 2}{2}$

Step 1.4

Combine the numerators over the common denominator.

$3 {(4 x)}^{- \frac{1}{6}} = \frac{- 17 + 10 \cdot 2}{2}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $10$ by $2$ .

$3 {(4 x)}^{- \frac{1}{6}} = \frac{- 17 + 20}{2}$

Step 1.5.2

Add $- 17$ and $20$ .

$3 {(4 x)}^{- \frac{1}{6}} = \frac{3}{2}$

Step 2

Raise each side of the equation to the power of $- 6$ to eliminate the fractional exponent on the left side.

${(3 {(4 x)}^{- \frac{1}{6}})}^{- 6} = {(\frac{3}{2})}^{- 6}$

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 ${(3 {(4 x)}^{- \frac{1}{6}})}^{- 6}$ .

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Step 3.1.1.1

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

${(3 \frac{1}{{(4 x)}^{\frac{1}{6}}})}^{- 6} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.2

Apply the product rule to $4 x$ .

${(3 \frac{1}{4^{\frac{1}{6}} x^{\frac{1}{6}}})}^{- 6} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.3

Combine $3$ and $\frac{1}{4^{\frac{1}{6}} x^{\frac{1}{6}}}$ .

${(\frac{3}{4^{\frac{1}{6}} x^{\frac{1}{6}}})}^{- 6} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.4

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

${(\frac{4^{\frac{1}{6}} x^{\frac{1}{6}}}{3})}^{6} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.1.5.1

Apply the product rule to $\frac{4^{\frac{1}{6}} x^{\frac{1}{6}}}{3}$ .

$\frac{{(4^{\frac{1}{6}} x^{\frac{1}{6}})}^{6}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.5.2

Apply the product rule to $4^{\frac{1}{6}} x^{\frac{1}{6}}$ .

$\frac{{(4^{\frac{1}{6}})}^{6} {(x^{\frac{1}{6}})}^{6}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.6

Simplify the numerator.

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Step 3.1.1.6.1

Multiply the exponents in ${(4^{\frac{1}{6}})}^{6}$ .

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Step 3.1.1.6.1.1

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

$\frac{4^{\frac{1}{6} \cdot 6} {(x^{\frac{1}{6}})}^{6}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.6.1.2

Cancel the common factor of $6$ .

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Step 3.1.1.6.1.2.1

Cancel the common factor.

$\frac{4^{\frac{1}{6} \cdot 6} {(x^{\frac{1}{6}})}^{6}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.6.1.2.2

Rewrite the expression.

$\frac{4^{1} {(x^{\frac{1}{6}})}^{6}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.6.2

Evaluate the exponent.

$\frac{4 {(x^{\frac{1}{6}})}^{6}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.6.3

Multiply the exponents in ${(x^{\frac{1}{6}})}^{6}$ .

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Step 3.1.1.6.3.1

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

$\frac{4 x^{\frac{1}{6} \cdot 6}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.6.3.2

Cancel the common factor of $6$ .

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Step 3.1.1.6.3.2.1

Cancel the common factor.

$\frac{4 x^{\frac{1}{6} \cdot 6}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.6.3.2.2

Rewrite the expression.

$\frac{4 x^{1}}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.6.4

Simplify.

$\frac{4 x}{3^{6}} = {(\frac{3}{2})}^{- 6}$

Step 3.1.1.7

Raise $3$ to the power of $6$ .

$\frac{4 x}{729} = {(\frac{3}{2})}^{- 6}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(\frac{3}{2})}^{- 6}$ .

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Step 3.2.1.1

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

$\frac{4 x}{729} = {(\frac{2}{3})}^{6}$

Step 3.2.1.2

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

$\frac{4 x}{729} = \frac{2^{6}}{3^{6}}$

Step 3.2.1.3

Raise $2$ to the power of $6$ .

$\frac{4 x}{729} = \frac{64}{3^{6}}$

Step 3.2.1.4

Raise $3$ to the power of $6$ .

$\frac{4 x}{729} = \frac{64}{729}$

Step 4

Solve for $x$ .

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Step 4.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$4 x = 64$

Step 4.2

Divide each term in $4 x = 64$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 x = 64$ by $4$ .

$\frac{4 x}{4} = \frac{64}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{64}{4}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{64}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $64$ by $4$ .

$x = 16$