Algebra Examples

$8 \sqrt[3]{2 x} = 4 x$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${(8 \sqrt[3]{2 x})}^{3} = {(4 x)}^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x}$ as ${(2 x)}^{\frac{1}{3}}$ .

${(8 {(2 x)}^{\frac{1}{3}})}^{3} = {(4 x)}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(8 {(2 x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Apply the product rule to $2 x$ .

${(8 (2^{\frac{1}{3}} x^{\frac{1}{3}}))}^{3} = {(4 x)}^{3}$

Step 2.2.1.2

Multiply $8 (2^{\frac{1}{3}} x^{\frac{1}{3}})$ .

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Step 2.2.1.2.1

Rewrite $8$ as $2^{3}$ .

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

Step 2.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.1.2.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.2.1.2.4

Combine $3$ and $\frac{3}{3}$ .

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

Step 2.2.1.2.5

Combine the numerators over the common denominator.

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

Step 2.2.1.2.6

Simplify the numerator.

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Step 2.2.1.2.6.1

Multiply $3$ by $3$ .

${(2^{\frac{1 + 9}{3}} x^{\frac{1}{3}})}^{3} = {(4 x)}^{3}$

Step 2.2.1.2.6.2

Add $1$ and $9$ .

${(2^{\frac{10}{3}} x^{\frac{1}{3}})}^{3} = {(4 x)}^{3}$

Step 2.2.1.3

Apply the product rule to $2^{\frac{10}{3}} x^{\frac{1}{3}}$ .

${(2^{\frac{10}{3}})}^{3} {(x^{\frac{1}{3}})}^{3} = {(4 x)}^{3}$

Step 2.2.1.4

Multiply the exponents in ${(2^{\frac{10}{3}})}^{3}$ .

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Step 2.2.1.4.1

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

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

Step 2.2.1.4.2

Cancel the common factor of $3$ .

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Step 2.2.1.4.2.1

Cancel the common factor.

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

Step 2.2.1.4.2.2

Rewrite the expression.

$2^{10} {(x^{\frac{1}{3}})}^{3} = {(4 x)}^{3}$

Step 2.2.1.5

Raise $2$ to the power of $10$ .

$1024 {(x^{\frac{1}{3}})}^{3} = {(4 x)}^{3}$

Step 2.2.1.6

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.6.1

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

$1024 x^{\frac{1}{3} \cdot 3} = {(4 x)}^{3}$

Step 2.2.1.6.2

Cancel the common factor of $3$ .

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Step 2.2.1.6.2.1

Cancel the common factor.

$1024 x^{\frac{1}{3} \cdot 3} = {(4 x)}^{3}$

Step 2.2.1.6.2.2

Rewrite the expression.

$1024 x^{1} = {(4 x)}^{3}$

Step 2.2.1.7

Simplify.

$1024 x = {(4 x)}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(4 x)}^{3}$ .

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Step 2.3.1.1

Apply the product rule to $4 x$ .

$1024 x = 4^{3} x^{3}$

Step 2.3.1.2

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

$1024 x = 64 x^{3}$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $64 x^{3}$ from both sides of the equation.

$1024 x - 64 x^{3} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $64 x$ out of $1024 x - 64 x^{3}$ .

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Step 3.2.1.1

Factor $64 x$ out of $1024 x$ .

$64 x (16) - 64 x^{3} = 0$

Step 3.2.1.2

Factor $64 x$ out of $- 64 x^{3}$ .

$64 x (16) + 64 x (- x^{2}) = 0$

Step 3.2.1.3

Factor $64 x$ out of $64 x (16) + 64 x (- x^{2})$ .

$64 x (16 - x^{2}) = 0$

Step 3.2.2

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

$64 x (4^{2} - x^{2}) = 0$

Step 3.2.3

Factor.

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Step 3.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x$ .

$64 x ((4 + x) (4 - x)) = 0$

Step 3.2.3.2

Remove unnecessary parentheses.

$64 x (4 + x) (4 - x) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$4 + x = 0$

$4 - x = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $4 + x$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $4 + x$ equal to $0$ .

$4 + x = 0$

Step 3.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.6

Set $4 - x$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $4 - x$ equal to $0$ .

$4 - x = 0$

Step 3.6.2

Solve $4 - x = 0$ for $x$ .

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Step 3.6.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 3.6.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 3.6.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 3.6.2.2.2

Simplify the left side.

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Step 3.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 3.6.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 3.6.2.2.3

Simplify the right side.

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Step 3.6.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 3.7

The final solution is all the values that make $64 x (4 + x) (4 - x) = 0$ true.

$x = 0, - 4, 4$