Algebra Examples

$x = \sqrt{8 x}$

Step 1

Subtract $\sqrt{8 x}$ from both sides of the equation.

$x - \sqrt{8 x} = 0$

Step 2

Simplify each term.

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Step 2.1

Rewrite $8 x$ as $2^{2} \cdot (2 x)$ .

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Step 2.1.1

Factor $4$ out of $8$ .

$x - \sqrt{4 (2) x} = 0$

Step 2.1.2

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

$x - \sqrt{2^{2} \cdot 2 x} = 0$

Step 2.1.3

Add parentheses.

$x - \sqrt{2^{2} \cdot (2 x)} = 0$

Step 2.2

Pull terms out from under the radical.

$x - (2 \sqrt{2 x}) = 0$

Step 2.3

Multiply $2$ by $- 1$ .

$x - 2 \sqrt{2 x} = 0$

Step 3

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

$x - 2 {(2 x)}^{\frac{1}{2}} = 0$

Step 4

Apply the product rule to $2 x$ .

$x - 2 (2^{\frac{1}{2}} x^{\frac{1}{2}}) = 0$

Step 5

Remove parentheses.

$x - 2 \cdot (2^{\frac{1}{2}} x^{\frac{1}{2}}) = 0$

Step 6

Multiply $- 2 \cdot 2^{\frac{1}{2}}$ .

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Step 6.1

Factor out negative.

$x - 2 \cdot 2^{\frac{1}{2}} x^{\frac{1}{2}} = 0$

Step 6.2

Raise $2$ to the power of $1$ .

$x - 2 \cdot 2^{\frac{1}{2}} x^{\frac{1}{2}} = 0$

Step 6.3

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

$x - 2^{1 + \frac{1}{2}} x^{\frac{1}{2}} = 0$

Step 6.4

Write $1$ as a fraction with a common denominator.

$x - 2^{\frac{2}{2} + \frac{1}{2}} x^{\frac{1}{2}} = 0$

Step 6.5

Combine the numerators over the common denominator.

$x - 2^{\frac{2 + 1}{2}} x^{\frac{1}{2}} = 0$

Step 6.6

Add $2$ and $1$ .

$x - 2^{\frac{3}{2}} x^{\frac{1}{2}} = 0$

Step 7

Factor $x^{\frac{1}{2}}$ out of $x - 2^{\frac{3}{2}} x^{\frac{1}{2}}$ .

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Step 7.1

Factor $x^{\frac{1}{2}}$ out of $x^{1}$ .

$x^{\frac{1}{2}} x^{\frac{1}{2}} - 2^{\frac{3}{2}} x^{\frac{1}{2}} = 0$

Step 7.2

Factor $x^{\frac{1}{2}}$ out of $- 2^{\frac{3}{2}} x^{\frac{1}{2}}$ .

$x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} (- 2^{\frac{3}{2}}) = 0$

Step 7.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} (- 2^{\frac{3}{2}})$ .

$x^{\frac{1}{2}} (x^{\frac{1}{2}} - 2^{\frac{3}{2}}) = 0$

Step 8

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

$x^{\frac{1}{2}} = 0$

$x^{\frac{1}{2}} - 2^{\frac{3}{2}} = 0$

Step 9

Set $x^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x^{\frac{1}{2}}$ equal to $0$ .

$x^{\frac{1}{2}} = 0$

Step 9.2

Solve $x^{\frac{1}{2}} = 0$ for $x$ .

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Step 9.2.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 9.2.2

Simplify the exponent.

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Step 9.2.2.1

Simplify the left side.

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Step 9.2.2.1.1

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

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Step 9.2.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 9.2.2.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 9.2.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 9.2.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 9.2.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 9.2.2.1.1.2

Simplify.

$x = 0^{2}$

Step 9.2.2.2

Simplify the right side.

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Step 9.2.2.2.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 10

Set $x^{\frac{1}{2}} - 2^{\frac{3}{2}}$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $x^{\frac{1}{2}} - 2^{\frac{3}{2}}$ equal to $0$ .

$x^{\frac{1}{2}} - 2^{\frac{3}{2}} = 0$

Step 10.2

Solve $x^{\frac{1}{2}} - 2^{\frac{3}{2}} = 0$ for $x$ .

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Step 10.2.1

Add $2^{\frac{3}{2}}$ to both sides of the equation.

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

Step 10.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 10.2.3

Simplify the exponent.

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Step 10.2.3.1

Simplify the left side.

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Step 10.2.3.1.1

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

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Step 10.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 10.2.3.1.1.1.1

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

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

Step 10.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 10.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 10.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 10.2.3.1.1.2

Simplify.

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

Step 10.2.3.2

Simplify the right side.

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Step 10.2.3.2.1

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

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Step 10.2.3.2.1.1

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

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Step 10.2.3.2.1.1.1

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

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

Step 10.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 10.2.3.2.1.1.2.1

Cancel the common factor.

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

Step 10.2.3.2.1.1.2.2

Rewrite the expression.

$x = 2^{3}$

Step 10.2.3.2.1.2

Raise $2$ to the power of $3$ .

$x = 8$

Step 11

The final solution is all the values that make $x^{\frac{1}{2}} (x^{\frac{1}{2}} - 2^{\frac{3}{2}}) = 0$ true.

$x = 0, 8$