Calculus Examples

$L (\sqrt[5]{0.95}) = 1 + \frac{x}{5}$

Step 1

Divide each term in $L (\sqrt[5]{0.95}) = 1 + \frac{x}{5}$ by $\sqrt[5]{0.95}$ .

$\frac{L \sqrt[5]{0.95}}{\sqrt[5]{0.95}} = \frac{1}{\sqrt[5]{0.95}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 2

Simplify the left side.

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Step 2.1

Cancel the common factor of $\sqrt[5]{0.95}$ .

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Step 2.1.1

Cancel the common factor.

$\frac{L \sqrt[5]{0.95}}{\sqrt[5]{0.95}} = \frac{1}{\sqrt[5]{0.95}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 2.1.2

Divide $L$ by $1$ .

$L = \frac{1}{\sqrt[5]{0.95}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3

Simplify the right side.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\frac{1}{\sqrt[5]{0.95}}$ by $\frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{4}}$ .

$L = \frac{1}{\sqrt[5]{0.95}} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{4}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2

Combine and simplify the denominator.

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Step 3.1.2.1

Multiply $\frac{1}{\sqrt[5]{0.95}}$ by $\frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{4}}$ .

$L = \frac{{\sqrt[5]{0.95}}^{4}}{\sqrt[5]{0.95} {\sqrt[5]{0.95}}^{4}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.2

Raise $\sqrt[5]{0.95}$ to the power of $1$ .

$L = \frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{1} {\sqrt[5]{0.95}}^{4}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.3

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

$L = \frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{1 + 4}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.4

Add $1$ and $4$ .

$L = \frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{5}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.5

Rewrite ${\sqrt[5]{0.95}}^{5}$ as $0.95$ .

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Step 3.1.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{0.95}$ as ${0.95}^{\frac{1}{5}}$ .

$L = \frac{{\sqrt[5]{0.95}}^{4}}{{({0.95}^{\frac{1}{5}})}^{5}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.5.2

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

$L = \frac{{\sqrt[5]{0.95}}^{4}}{{0.95}^{\frac{1}{5} \cdot 5}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.5.3

Combine $\frac{1}{5}$ and $5$ .

$L = \frac{{\sqrt[5]{0.95}}^{4}}{{0.95}^{\frac{5}{5}}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.5.4

Cancel the common factor of $5$ .

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Step 3.1.2.5.4.1

Cancel the common factor.

$L = \frac{{\sqrt[5]{0.95}}^{4}}{{0.95}^{\frac{5}{5}}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.5.4.2

Rewrite the expression.

$L = \frac{{\sqrt[5]{0.95}}^{4}}{{0.95}^{1}} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.2.5.5

Evaluate the exponent.

$L = \frac{{\sqrt[5]{0.95}}^{4}}{0.95} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.3

Simplify the numerator.

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Step 3.1.3.1

Rewrite ${\sqrt[5]{0.95}}^{4}$ as $\sqrt[5]{{0.95}^{4}}$ .

$L = \frac{\sqrt[5]{{0.95}^{4}}}{0.95} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.3.2

Raise $0.95$ to the power of $4$ .

$L = \frac{\sqrt[5]{0.81450625}}{0.95} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.4

Evaluate the root.

$L = \frac{0.95979588}{0.95} + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.5

Divide $0.95979588$ by $0.95$ .

$L = 1.01031145 + \frac{\frac{x}{5}}{\sqrt[5]{0.95}}$

Step 3.1.6

Multiply the numerator by the reciprocal of the denominator.

$L = 1.01031145 + \frac{x}{5} \cdot \frac{1}{\sqrt[5]{0.95}}$

Step 3.1.7

Multiply $\frac{1}{\sqrt[5]{0.95}}$ by $\frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{4}}$ .

$L = 1.01031145 + \frac{x}{5} (\frac{1}{\sqrt[5]{0.95}} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{4}})$

Step 3.1.8

Combine and simplify the denominator.

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Step 3.1.8.1

Multiply $\frac{1}{\sqrt[5]{0.95}}$ by $\frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{4}}$ .

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{\sqrt[5]{0.95} {\sqrt[5]{0.95}}^{4}}$

Step 3.1.8.2

Raise $\sqrt[5]{0.95}$ to the power of $1$ .

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{1} {\sqrt[5]{0.95}}^{4}}$

Step 3.1.8.3

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

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{1 + 4}}$

Step 3.1.8.4

Add $1$ and $4$ .

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{\sqrt[5]{0.95}}^{5}}$

Step 3.1.8.5

Rewrite ${\sqrt[5]{0.95}}^{5}$ as $0.95$ .

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Step 3.1.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{0.95}$ as ${0.95}^{\frac{1}{5}}$ .

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{({0.95}^{\frac{1}{5}})}^{5}}$

Step 3.1.8.5.2

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

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{0.95}^{\frac{1}{5} \cdot 5}}$

Step 3.1.8.5.3

Combine $\frac{1}{5}$ and $5$ .

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{0.95}^{\frac{5}{5}}}$

Step 3.1.8.5.4

Cancel the common factor of $5$ .

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Step 3.1.8.5.4.1

Cancel the common factor.

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{0.95}^{\frac{5}{5}}}$

Step 3.1.8.5.4.2

Rewrite the expression.

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{{0.95}^{1}}$

Step 3.1.8.5.5

Evaluate the exponent.

$L = 1.01031145 + \frac{x}{5} \cdot \frac{{\sqrt[5]{0.95}}^{4}}{0.95}$

Step 3.1.9

Combine.

$L = 1.01031145 + \frac{x {\sqrt[5]{0.95}}^{4}}{5 \cdot 0.95}$

Step 3.1.10

Simplify the numerator.

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Step 3.1.10.1

Rewrite ${\sqrt[5]{0.95}}^{4}$ as $\sqrt[5]{{0.95}^{4}}$ .

$L = 1.01031145 + \frac{x \sqrt[5]{{0.95}^{4}}}{5 \cdot 0.95}$

Step 3.1.10.2

Raise $0.95$ to the power of $4$ .

$L = 1.01031145 + \frac{x \sqrt[5]{0.81450625}}{5 \cdot 0.95}$

Step 3.1.11

Multiply $5$ by $0.95$ .

$L = 1.01031145 + \frac{x \sqrt[5]{0.81450625}}{4.75}$

Step 3.1.12

Factor $\sqrt[5]{0.81450625}$ out of $x \sqrt[5]{0.81450625}$ .

$L = 1.01031145 + \frac{\sqrt[5]{0.81450625} x}{4.75}$

Step 3.1.13

Factor $4.75$ out of $4.75$ .

$L = 1.01031145 + \frac{\sqrt[5]{0.81450625} x}{4.75 (1)}$

Step 3.1.14

Separate fractions.

$L = 1.01031145 + \frac{\sqrt[5]{0.81450625}}{4.75} \cdot \frac{x}{1}$

Step 3.1.15

Evaluate the root.

$L = 1.01031145 + \frac{0.95979588}{4.75} \cdot \frac{x}{1}$

Step 3.1.16

Divide $0.95979588$ by $4.75$ .

$L = 1.01031145 + 0.20206229 \frac{x}{1}$

Step 3.1.17

Divide $x$ by $1$ .

$L = 1.01031145 + 0.20206229 x$