Calculus Examples

$f^{4} (x) = \frac{6}{\sqrt{x}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{6}{\sqrt{x}} = f^{4} x$

Step 2

Cross multiply.

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Step 2.1

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.

$f^{4} x \cdot (\sqrt{x}) = 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Reorder factors in $f^{4} x (\sqrt{x})$ .

$f^{4} x \sqrt{x} = 6$

Step 3

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

${(f^{4} x \sqrt{x})}^{2} = 6^{2}$

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(f^{4} x \cdot x^{\frac{1}{2}})}^{2} = 6^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

${(f^{4} (x^{\frac{1}{2}} x))}^{2} = 6^{2}$

Step 4.2.1.1.2

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

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Step 4.2.1.1.2.1

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

${(f^{4} (x^{\frac{1}{2}} x^{1}))}^{2} = 6^{2}$

Step 4.2.1.1.2.2

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

${(f^{4} x^{\frac{1}{2} + 1})}^{2} = 6^{2}$

Step 4.2.1.1.3

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

${(f^{4} x^{\frac{1}{2} + \frac{2}{2}})}^{2} = 6^{2}$

Step 4.2.1.1.4

Combine the numerators over the common denominator.

${(f^{4} x^{\frac{1 + 2}{2}})}^{2} = 6^{2}$

Step 4.2.1.1.5

Add $1$ and $2$ .

${(f^{4} x^{\frac{3}{2}})}^{2} = 6^{2}$

Step 4.2.1.2

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

${(f^{4})}^{2} {(x^{\frac{3}{2}})}^{2} = 6^{2}$

Step 4.2.1.3

Multiply the exponents in ${(f^{4})}^{2}$ .

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Step 4.2.1.3.1

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

$f^{4 \cdot 2} {(x^{\frac{3}{2}})}^{2} = 6^{2}$

Step 4.2.1.3.2

Multiply $4$ by $2$ .

$f^{8} {(x^{\frac{3}{2}})}^{2} = 6^{2}$

Step 4.2.1.4

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

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Step 4.2.1.4.1

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

$f^{8} x^{\frac{3}{2} \cdot 2} = 6^{2}$

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.

$f^{8} x^{\frac{3}{2} \cdot 2} = 6^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

$f^{8} x^{3} = 6^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Raise $6$ to the power of $2$ .

$f^{8} x^{3} = 36$

Step 5

Solve for $x$ .

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Step 5.1

Divide each term in $f^{8} x^{3} = 36$ by $f^{8}$ and simplify.

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Step 5.1.1

Divide each term in $f^{8} x^{3} = 36$ by $f^{8}$ .

$\frac{f^{8} x^{3}}{f^{8}} = \frac{36}{f^{8}}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $f^{8}$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{f^{8} x^{3}}{f^{8}} = \frac{36}{f^{8}}$

Step 5.1.2.1.2

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

$x^{3} = \frac{36}{f^{8}}$

Step 5.2

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

$x = \sqrt[3]{\frac{36}{f^{8}}}$

Step 5.3

Simplify $\sqrt[3]{\frac{36}{f^{8}}}$ .

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Step 5.3.1

Rewrite $\frac{36}{f^{8}}$ as ${(\frac{1}{f^{2}})}^{3} \frac{36}{f^{2}}$ .

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Step 5.3.1.1

Factor the perfect power $1^{3}$ out of $36$ .

$x = \sqrt[3]{\frac{1^{3} \cdot 36}{f^{8}}}$

Step 5.3.1.2

Factor the perfect power ${(f^{2})}^{3}$ out of $f^{8}$ .

$x = \sqrt[3]{\frac{1^{3} \cdot 36}{{(f^{2})}^{3} f^{2}}}$

Step 5.3.1.3

Rearrange the fraction $\frac{1^{3} \cdot 36}{{(f^{2})}^{3} f^{2}}$ .

$x = \sqrt[3]{{(\frac{1}{f^{2}})}^{3} \frac{36}{f^{2}}}$

Step 5.3.2

Pull terms out from under the radical.

$x = \frac{1}{f^{2}} \sqrt[3]{\frac{36}{f^{2}}}$

Step 5.3.3

Rewrite $\sqrt[3]{\frac{36}{f^{2}}}$ as $\frac{\sqrt[3]{36}}{\sqrt[3]{f^{2}}}$ .

$x = \frac{1}{f^{2}} \cdot \frac{\sqrt[3]{36}}{\sqrt[3]{f^{2}}}$

Step 5.3.4

Combine.

$x = \frac{1 \sqrt[3]{36}}{f^{2} \sqrt[3]{f^{2}}}$

Step 5.3.5

Multiply $\sqrt[3]{36}$ by $1$ .

$x = \frac{\sqrt[3]{36}}{f^{2} \sqrt[3]{f^{2}}}$

Step 5.3.6

Multiply $\frac{\sqrt[3]{36}}{f^{2} \sqrt[3]{f^{2}}}$ by $\frac{{\sqrt[3]{f^{2}}}^{2}}{{\sqrt[3]{f^{2}}}^{2}}$ .

$x = \frac{\sqrt[3]{36}}{f^{2} \sqrt[3]{f^{2}}} \cdot \frac{{\sqrt[3]{f^{2}}}^{2}}{{\sqrt[3]{f^{2}}}^{2}}$

Step 5.3.7

Combine and simplify the denominator.

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Step 5.3.7.1

Multiply $\frac{\sqrt[3]{36}}{f^{2} \sqrt[3]{f^{2}}}$ by $\frac{{\sqrt[3]{f^{2}}}^{2}}{{\sqrt[3]{f^{2}}}^{2}}$ .

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} \sqrt[3]{f^{2}} {\sqrt[3]{f^{2}}}^{2}}$

Step 5.3.7.2

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

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} (\sqrt[3]{f^{2}} {\sqrt[3]{f^{2}}}^{2})}$

Step 5.3.7.3

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

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} ({\sqrt[3]{f^{2}}}^{1} {\sqrt[3]{f^{2}}}^{2})}$

Step 5.3.7.4

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

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} {\sqrt[3]{f^{2}}}^{1 + 2}}$

Step 5.3.7.5

Add $1$ and $2$ .

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} {\sqrt[3]{f^{2}}}^{3}}$

Step 5.3.7.6

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

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Step 5.3.7.6.1

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

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} {(f^{\frac{2}{3}})}^{3}}$

Step 5.3.7.6.2

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

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} f^{\frac{2}{3} \cdot 3}}$

Step 5.3.7.6.3

Combine $\frac{2}{3}$ and $3$ .

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} f^{\frac{2 \cdot 3}{3}}}$

Step 5.3.7.6.4

Multiply $2$ by $3$ .

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} f^{\frac{6}{3}}}$

Step 5.3.7.6.5

Cancel the common factor of $6$ and $3$ .

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Step 5.3.7.6.5.1

Factor $3$ out of $6$ .

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} f^{\frac{3 \cdot 2}{3}}}$

Step 5.3.7.6.5.2

Cancel the common factors.

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Step 5.3.7.6.5.2.1

Factor $3$ out of $3$ .

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} f^{\frac{3 \cdot 2}{3 (1)}}}$

Step 5.3.7.6.5.2.2

Cancel the common factor.

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} f^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Step 5.3.7.6.5.2.3

Rewrite the expression.

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} f^{\frac{2}{1}}}$

Step 5.3.7.6.5.2.4

Divide $2$ by $1$ .

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2} f^{2}}$

Step 5.3.8

Multiply $f^{2}$ by $f^{2}$ by adding the exponents.

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Step 5.3.8.1

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

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{2 + 2}}$

Step 5.3.8.2

Add $2$ and $2$ .

$x = \frac{\sqrt[3]{36} {\sqrt[3]{f^{2}}}^{2}}{f^{4}}$

Step 5.3.9

Simplify the numerator.

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Step 5.3.9.1

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

$x = \frac{\sqrt[3]{36} \sqrt[3]{{(f^{2})}^{2}}}{f^{4}}$

Step 5.3.9.2

Multiply the exponents in ${(f^{2})}^{2}$ .

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Step 5.3.9.2.1

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

$x = \frac{\sqrt[3]{36} \sqrt[3]{f^{2 \cdot 2}}}{f^{4}}$

Step 5.3.9.2.2

Multiply $2$ by $2$ .

$x = \frac{\sqrt[3]{36} \sqrt[3]{f^{4}}}{f^{4}}$

Step 5.3.9.3

Factor out $f^{3}$ .

$x = \frac{\sqrt[3]{36} \sqrt[3]{f^{3} f}}{f^{4}}$

Step 5.3.9.4

Pull terms out from under the radical.

$x = \frac{\sqrt[3]{36} f \sqrt[3]{f}}{f^{4}}$

Step 5.3.9.5

Combine using the product rule for radicals.

$x = \frac{f \sqrt[3]{36 f}}{f^{4}}$

Step 5.3.10

Cancel the common factor of $f$ and $f^{4}$ .

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Step 5.3.10.1

Factor $f$ out of $f \sqrt[3]{36 f}$ .

$x = \frac{f (\sqrt[3]{36 f})}{f^{4}}$

Step 5.3.10.2

Cancel the common factors.

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Step 5.3.10.2.1

Factor $f$ out of $f^{4}$ .

$x = \frac{f (\sqrt[3]{36 f})}{f \cdot f^{3}}$

Step 5.3.10.2.2

Cancel the common factor.

$x = \frac{f \sqrt[3]{36 f}}{f \cdot f^{3}}$

Step 5.3.10.2.3

Rewrite the expression.

$x = \frac{\sqrt[3]{36 f}}{f^{3}}$

Step 6

To rewrite as a function of $f$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $f$ is on the other side.

$f (f) = \frac{\sqrt[3]{36 f}}{f^{3}}$