Algebra Examples

$\frac{1}{4} \sqrt{32 x} = 1$

Step 1

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} \cdot \sqrt{32 x}) = 4 \cdot 1$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $4 (\frac{1}{4} \cdot \sqrt{32 x})$ .

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Step 2.1.1.1

Rewrite $32 x$ as ${(2^{2})}^{2} \cdot (2 x)$ .

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Step 2.1.1.1.1

Factor $16$ out of $32$ .

$4 (\frac{1}{4} \cdot \sqrt{16 (2) x}) = 4 \cdot 1$

Step 2.1.1.1.2

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

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

Step 2.1.1.1.3

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

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

Step 2.1.1.1.4

Add parentheses.

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

Step 2.1.1.2

Pull terms out from under the radical.

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Cancel the common factor of $4$ .

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Step 2.1.1.4.1

Factor $4$ out of $4 \sqrt{2 x}$ .

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

Step 2.1.1.4.2

Cancel the common factor.

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

Step 2.1.1.4.3

Rewrite the expression.

$4 \sqrt{2 x} = 4 \cdot 1$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $4$ by $1$ .

$4 \sqrt{2 x} = 4$

Step 3

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

${(4 \sqrt{2 x})}^{2} = 4^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(4 {(2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $2 x$ .

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

Step 4.2.1.2

Multiply $4 (2^{\frac{1}{2}} x^{\frac{1}{2}})$ .

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Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

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

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

Step 4.2.1.2.3

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

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

Step 4.2.1.2.4

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

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

Step 4.2.1.2.5

Combine the numerators over the common denominator.

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

Step 4.2.1.2.6

Simplify the numerator.

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Step 4.2.1.2.6.1

Multiply $2$ by $2$ .

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

Step 4.2.1.2.6.2

Add $1$ and $4$ .

${(2^{\frac{5}{2}} x^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.3

Apply the product rule to $2^{\frac{5}{2}} x^{\frac{1}{2}}$ .

${(2^{\frac{5}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.4

Multiply the exponents in ${(2^{\frac{5}{2}})}^{2}$ .

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Step 4.2.1.4.1

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

$2^{\frac{5}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

$2^{\frac{5}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

$2^{5} {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.5

Raise $2$ to the power of $5$ .

$32 {(x^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4.2.1.6

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

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Step 4.2.1.6.1

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

$32 x^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 4.2.1.6.2

Cancel the common factor of $2$ .

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Step 4.2.1.6.2.1

Cancel the common factor.

$32 x^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 4.2.1.6.2.2

Rewrite the expression.

$32 x^{1} = 4^{2}$

Step 4.2.1.7

Simplify.

$32 x = 4^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Raise $4$ to the power of $2$ .

$32 x = 16$

Step 5

Divide each term in $32 x = 16$ by $32$ and simplify.

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Step 5.1

Divide each term in $32 x = 16$ by $32$ .

$\frac{32 x}{32} = \frac{16}{32}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $32$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{32 x}{32} = \frac{16}{32}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{32}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $16$ and $32$ .

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Step 5.3.1.1

Factor $16$ out of $16$ .

$x = \frac{16 (1)}{32}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $16$ out of $32$ .

$x = \frac{16 \cdot 1}{16 \cdot 2}$

Step 5.3.1.2.2

Cancel the common factor.

$x = \frac{16 \cdot 1}{16 \cdot 2}$

Step 5.3.1.2.3

Rewrite the expression.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$