Calculus Examples

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

Step 1

Cross multiply.

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Step 1.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $\sqrt{2 x} \cdot (2 \sqrt{2 x^{2}})$ .

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Step 1.2.1.1

Simplify the expression.

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Step 1.2.1.1.1

Rewrite using the commutative property of multiplication.

$2 \sqrt{2 x} \sqrt{2 x^{2}} = 1$

Step 1.2.1.1.2

Reorder $2$ and $x^{2}$ .

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

Step 1.2.1.2

Pull terms out from under the radical.

$2 \sqrt{2 x} (x \sqrt{2}) = 1$

Step 1.2.1.3

Multiply $2 \sqrt{2 x} (x \sqrt{2})$ .

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Step 1.2.1.3.1

Combine using the product rule for radicals.

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

Step 1.2.1.3.2

Multiply $2$ by $2$ .

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

Step 1.2.1.4

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

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

Step 1.2.1.5

Pull terms out from under the radical.

$2 (x (2 \sqrt{x})) = 1$

Step 1.2.1.6

Simplify the expression.

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Step 1.2.1.6.1

Rewrite using the commutative property of multiplication.

$2 (2 x \sqrt{x}) = 1$

Step 1.2.1.6.2

Multiply $2$ by $2$ .

$4 x \sqrt{x} = 1$

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 3.2.1.1.1

Move $x^{\frac{1}{2}}$ .

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

Step 3.2.1.1.2

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

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Step 3.2.1.1.2.1

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

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

Step 3.2.1.1.2.2

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

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

Step 3.2.1.1.3

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

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

Step 3.2.1.1.4

Combine the numerators over the common denominator.

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

Step 3.2.1.1.5

Add $1$ and $2$ .

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

Step 3.2.1.2

Apply the product rule to $4 x^{\frac{3}{2}}$ .

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

Step 3.2.1.3

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

$16 {(x^{\frac{3}{2}})}^{2} = 1^{2}$

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

$16 x^{\frac{3}{2} \cdot 2} = 1^{2}$

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

$16 x^{\frac{3}{2} \cdot 2} = 1^{2}$

Step 3.2.1.4.2.2

Rewrite the expression.

$16 x^{3} = 1^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

One to any power is one.

$16 x^{3} = 1$

Step 4

Solve for $x$ .

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Step 4.1

Divide each term in $16 x^{3} = 1$ by $16$ and simplify.

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Step 4.1.1

Divide each term in $16 x^{3} = 1$ by $16$ .

$\frac{16 x^{3}}{16} = \frac{1}{16}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $16$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{16 x^{3}}{16} = \frac{1}{16}$

Step 4.1.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{1}{16}$

Step 4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{\frac{1}{16}}$

Step 4.3

Simplify $\sqrt[3]{\frac{1}{16}}$ .

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Step 4.3.1

Rewrite $\sqrt[3]{\frac{1}{16}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{16}}$ .

$x = \frac{\sqrt[3]{1}}{\sqrt[3]{16}}$

Step 4.3.2

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt[3]{16}}$

Step 4.3.3

Simplify the denominator.

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Step 4.3.3.1

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 4.3.3.1.1

Factor $8$ out of $16$ .

$x = \frac{1}{\sqrt[3]{8 (2)}}$

Step 4.3.3.1.2

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

$x = \frac{1}{\sqrt[3]{2^{3} \cdot 2}}$

Step 4.3.3.2

Pull terms out from under the radical.

$x = \frac{1}{2 \sqrt[3]{2}}$

Step 4.3.4

Multiply $\frac{1}{2 \sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{1}{2 \sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 4.3.5

Combine and simplify the denominator.

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Step 4.3.5.1

Multiply $\frac{1}{2 \sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{2 \sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 4.3.5.2

Move $\sqrt[3]{2}$ .

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

Step 4.3.5.3

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 4.3.5.4

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

$x = \frac{{\sqrt[3]{2}}^{2}}{2 {\sqrt[3]{2}}^{1 + 2}}$

Step 4.3.5.5

Add $1$ and $2$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{2 {\sqrt[3]{2}}^{3}}$

Step 4.3.5.6

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 4.3.5.6.1

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

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

Step 4.3.5.6.2

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

$x = \frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2^{\frac{1}{3} \cdot 3}}$

Step 4.3.5.6.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2^{\frac{3}{3}}}$

Step 4.3.5.6.4

Cancel the common factor of $3$ .

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Step 4.3.5.6.4.1

Cancel the common factor.

$x = \frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2^{\frac{3}{3}}}$

Step 4.3.5.6.4.2

Rewrite the expression.

$x = \frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2^{1}}$

Step 4.3.5.6.5

Evaluate the exponent.

$x = \frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2}$

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$x = \frac{\sqrt[3]{2^{2}}}{2 \cdot 2}$

Step 4.3.6.2

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

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

Step 4.3.7

Multiply $2$ by $2$ .

$x = \frac{\sqrt[3]{4}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[3]{4}}{4}$

Decimal Form:

$x = 0.39685026 \dots$