Algebra Examples

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

Step 1

Apply the product rule to $8 x$ .

$8^{\frac{1}{2}} x^{\frac{1}{2}} = x$

Step 2

Subtract $x$ from both sides of the equation.

$8^{\frac{1}{2}} x^{\frac{1}{2}} - x = 0$

Step 3

Factor $x^{\frac{1}{2}}$ out of $8^{\frac{1}{2}} x^{\frac{1}{2}} - x$ .

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Step 3.1

Factor $x^{\frac{1}{2}}$ out of $8^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$x^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} - x = 0$

Step 3.2

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

$x^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} + x^{\frac{1}{2}} (- x^{\frac{1}{2}}) = 0$

Step 3.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} + x^{\frac{1}{2}} (- x^{\frac{1}{2}})$ .

$x^{\frac{1}{2}} (8^{\frac{1}{2}} - x^{\frac{1}{2}}) = 0$

Step 4

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

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

$8^{\frac{1}{2}} - x^{\frac{1}{2}} = 0$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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 5.2.2

Simplify the exponent.

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Step 5.2.2.1

Simplify the left side.

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Step 5.2.2.1.1

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

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Step 5.2.2.1.1.1

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

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Step 5.2.2.1.1.1.1

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

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

Step 5.2.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.2.1.1.1.2.1

Cancel the common factor.

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

Step 5.2.2.1.1.1.2.2

Rewrite the expression.

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

Step 5.2.2.1.1.2

Simplify.

$x = 0^{2}$

Step 5.2.2.2

Simplify the right side.

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Step 5.2.2.2.1

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

$x = 0$

Step 6

Set $8^{\frac{1}{2}} - x^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $8^{\frac{1}{2}} - x^{\frac{1}{2}}$ equal to $0$ .

$8^{\frac{1}{2}} - x^{\frac{1}{2}} = 0$

Step 6.2

Solve $8^{\frac{1}{2}} - x^{\frac{1}{2}} = 0$ for $x$ .

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Step 6.2.1

Subtract $8^{\frac{1}{2}}$ from both sides of the equation.

$- x^{\frac{1}{2}} = - 8^{\frac{1}{2}}$

Step 6.2.2

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

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

Step 6.2.3

Simplify the exponent.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

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

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Step 6.2.3.1.1.1

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

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

Step 6.2.3.1.1.2

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

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

Step 6.2.3.1.1.3

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

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

Step 6.2.3.1.1.4

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

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Step 6.2.3.1.1.4.1

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

$x^{\frac{1}{2} \cdot 2} = {(- 8^{\frac{1}{2}})}^{2}$

Step 6.2.3.1.1.4.2

Cancel the common factor of $2$ .

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Step 6.2.3.1.1.4.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(- 8^{\frac{1}{2}})}^{2}$

Step 6.2.3.1.1.4.2.2

Rewrite the expression.

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

Step 6.2.3.1.1.5

Simplify.

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

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

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

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Step 6.2.3.2.1.1

Apply the product rule to $- 8^{\frac{1}{2}}$ .

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

Step 6.2.3.2.1.2

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

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

Step 6.2.3.2.1.3

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

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

Step 6.2.3.2.1.4

Multiply the exponents in ${(8^{\frac{1}{2}})}^{2}$ .

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Step 6.2.3.2.1.4.1

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

$x = 8^{\frac{1}{2} \cdot 2}$

Step 6.2.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 6.2.3.2.1.4.2.1

Cancel the common factor.

$x = 8^{\frac{1}{2} \cdot 2}$

Step 6.2.3.2.1.4.2.2

Rewrite the expression.

$x = 8^{1}$

Step 6.2.3.2.1.5

Evaluate the exponent.

$x = 8$

Step 7

The final solution is all the values that make $x^{\frac{1}{2}} (8^{\frac{1}{2}} - x^{\frac{1}{2}}) = 0$ true.

$x = 0, 8$