Calculus Examples

$y = x + 9 x^{2}$ , $[- 2, 4]$

Step 1

The Root Mean Square (RMS) of a function $f$ over a specified interval $[a, b]$ is the square root of the arithmetic mean (average) of the squares of the original values.

$f_{rms} = \sqrt{\frac{1}{b - a} \cdot \int_{a}^{b} f {(x)}^{2} d x}$

Step 2

Substitute the actual values into the formula for the root mean square of a function.

$f_{rms} = \sqrt{\frac{1}{4 + 2} \cdot (\int_{- 2}^{4} {(x + 9 x^{2})}^{2} d x)}$

Step 3

Evaluate the integral.

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Step 3.1

Expand ${(x + 9 x^{2})}^{2}$ .

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Step 3.1.1

Rewrite ${(x + 9 x^{2})}^{2}$ as $(x + 9 x^{2}) (x + 9 x^{2})$ .

$\int_{- 2}^{4} (x + 9 x^{2}) (x + 9 x^{2}) d x$

Step 3.1.2

Apply the distributive property.

$\int_{- 2}^{4} x (x + 9 x^{2}) + 9 x^{2} (x + 9 x^{2}) d x$

Step 3.1.3

Apply the distributive property.

$\int_{- 2}^{4} x \cdot x + x (9 x^{2}) + 9 x^{2} (x + 9 x^{2}) d x$

Step 3.1.4

Apply the distributive property.

$\int_{- 2}^{4} x \cdot x + x (9 x^{2}) + 9 x^{2} x + 9 x^{2} (9 x^{2}) d x$

Step 3.1.5

Reorder $x$ and $9$ .

$\int_{- 2}^{4} x \cdot x + 9 \cdot x \cdot x^{2} + 9 x^{2} x + 9 x^{2} (9 x^{2}) d x$

Step 3.1.6

Move $x^{2}$ .

$\int_{- 2}^{4} x \cdot x + 9 \cdot x \cdot x^{2} + 9 x^{2} x + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.7

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

$\int_{- 2}^{4} x^{1} x + 9 x \cdot x^{2} + 9 x^{2} x + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.8

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

$\int_{- 2}^{4} x^{1} x^{1} + 9 x \cdot x^{2} + 9 x^{2} x + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.9

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

$\int_{- 2}^{4} x^{1 + 1} + 9 x \cdot x^{2} + 9 x^{2} x + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.10

Add $1$ and $1$ .

$\int_{- 2}^{4} x^{2} + 9 x \cdot x^{2} + 9 x^{2} x + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.11

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

$\int_{- 2}^{4} x^{2} + 9 (x^{1} x^{2}) + 9 x^{2} x + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.12

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

$\int_{- 2}^{4} x^{2} + 9 x^{1 + 2} + 9 x^{2} x + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.13

Add $1$ and $2$ .

$\int_{- 2}^{4} x^{2} + 9 x^{3} + 9 x^{2} x + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.14

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

$\int_{- 2}^{4} x^{2} + 9 x^{3} + 9 (x^{2} x^{1}) + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.15

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

$\int_{- 2}^{4} x^{2} + 9 x^{3} + 9 x^{2 + 1} + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.16

Add $2$ and $1$ .

$\int_{- 2}^{4} x^{2} + 9 x^{3} + 9 x^{3} + 9 \cdot 9 x^{2} x^{2} d x$

Step 3.1.17

Multiply $9$ by $9$ .

$\int_{- 2}^{4} x^{2} + 9 x^{3} + 9 x^{3} + 81 x^{2} x^{2} d x$

Step 3.1.18

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

$\int_{- 2}^{4} x^{2} + 9 x^{3} + 9 x^{3} + 81 x^{2 + 2} d x$

Step 3.1.19

Add $2$ and $2$ .

$\int_{- 2}^{4} x^{2} + 9 x^{3} + 9 x^{3} + 81 x^{4} d x$

Step 3.1.20

Add $9 x^{3}$ and $9 x^{3}$ .

$\int_{- 2}^{4} x^{2} + 18 x^{3} + 81 x^{4} d x$

Step 3.1.21

Reorder $18 x^{3}$ and $81 x^{4}$ .

$\int_{- 2}^{4} x^{2} + 81 x^{4} + 18 x^{3} d x$

Step 3.1.22

Move $x^{2}$ .

$\int_{- 2}^{4} 81 x^{4} + 18 x^{3} + x^{2} d x$

Step 3.2

Split the single integral into multiple integrals.

$\int_{- 2}^{4} 81 x^{4} d x + \int_{- 2}^{4} 18 x^{3} d x + \int_{- 2}^{4} x^{2} d x$

Step 3.3

Since $81$ is constant with respect to $x$ , move $81$ out of the integral.

$81 \int_{- 2}^{4} x^{4} d x + \int_{- 2}^{4} 18 x^{3} d x + \int_{- 2}^{4} x^{2} d x$

Step 3.4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$81 (\frac{1}{5} x^{5}]_{- 2}^{4}) + \int_{- 2}^{4} 18 x^{3} d x + \int_{- 2}^{4} x^{2} d x$

Step 3.5

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

$81 (\frac{x^{5}}{5}]_{- 2}^{4}) + \int_{- 2}^{4} 18 x^{3} d x + \int_{- 2}^{4} x^{2} d x$

Step 3.6

Since $18$ is constant with respect to $x$ , move $18$ out of the integral.

$81 (\frac{x^{5}}{5}]_{- 2}^{4}) + 18 \int_{- 2}^{4} x^{3} d x + \int_{- 2}^{4} x^{2} d x$

Step 3.7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$81 (\frac{x^{5}}{5}]_{- 2}^{4}) + 18 (\frac{1}{4} x^{4}]_{- 2}^{4}) + \int_{- 2}^{4} x^{2} d x$

Step 3.8

Combine $\frac{1}{4}$ and $x^{4}$ .

$81 (\frac{x^{5}}{5}]_{- 2}^{4}) + 18 (\frac{x^{4}}{4}]_{- 2}^{4}) + \int_{- 2}^{4} x^{2} d x$

Step 3.9

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$81 (\frac{x^{5}}{5}]_{- 2}^{4}) + 18 (\frac{x^{4}}{4}]_{- 2}^{4}) + \frac{1}{3} x^{3}]_{- 2}^{4}$

Step 3.10

Substitute and simplify.

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Step 3.10.1

Evaluate $\frac{x^{5}}{5}$ at $4$ and at $- 2$ .

$81 ((\frac{4^{5}}{5}) - \frac{{(- 2)}^{5}}{5}) + 18 (\frac{x^{4}}{4}]_{- 2}^{4}) + \frac{1}{3} x^{3}]_{- 2}^{4}$

Step 3.10.2

Evaluate $\frac{x^{4}}{4}$ at $4$ and at $- 2$ .

$81 (\frac{4^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + \frac{1}{3} x^{3}]_{- 2}^{4}$

Step 3.10.3

Evaluate $\frac{1}{3} x^{3}$ at $4$ and at $- 2$ .

$81 (\frac{4^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4

Simplify.

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Step 3.10.4.1

Raise $4$ to the power of $5$ .

$81 (\frac{1024}{5} - \frac{{(- 2)}^{5}}{5}) + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.2

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

$81 (\frac{1024}{5} - \frac{- 32}{5}) + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.3

Move the negative in front of the fraction.

$81 (\frac{1024}{5} - - \frac{32}{5}) + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.4

Multiply $- 1$ by $- 1$ .

$81 (\frac{1024}{5} + 1 (\frac{32}{5})) + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.5

Multiply $\frac{32}{5}$ by $1$ .

$81 (\frac{1024}{5} + \frac{32}{5}) + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.6

Combine the numerators over the common denominator.

$81 \frac{1024 + 32}{5} + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.7

Add $1024$ and $32$ .

$81 (\frac{1056}{5}) + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.8

Combine $81$ and $\frac{1056}{5}$ .

$\frac{81 \cdot 1056}{5} + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.9

Multiply $81$ by $1056$ .

$\frac{85536}{5} + 18 (\frac{4^{4}}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.10

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

$\frac{85536}{5} + 18 (\frac{256}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.11

Cancel the common factor of $256$ and $4$ .

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Step 3.10.4.11.1

Factor $4$ out of $256$ .

$\frac{85536}{5} + 18 (\frac{4 \cdot 64}{4} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.11.2

Cancel the common factors.

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Step 3.10.4.11.2.1

Factor $4$ out of $4$ .

$\frac{85536}{5} + 18 (\frac{4 \cdot 64}{4 (1)} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.11.2.2

Cancel the common factor.

$\frac{85536}{5} + 18 (\frac{4 \cdot 64}{4 \cdot 1} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.11.2.3

Rewrite the expression.

$\frac{85536}{5} + 18 (\frac{64}{1} - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.11.2.4

Divide $64$ by $1$ .

$\frac{85536}{5} + 18 (64 - \frac{{(- 2)}^{4}}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.12

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

$\frac{85536}{5} + 18 (64 - \frac{16}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.13

Cancel the common factor of $16$ and $4$ .

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Step 3.10.4.13.1

Factor $4$ out of $16$ .

$\frac{85536}{5} + 18 (64 - \frac{4 \cdot 4}{4}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.13.2

Cancel the common factors.

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Step 3.10.4.13.2.1

Factor $4$ out of $4$ .

$\frac{85536}{5} + 18 (64 - \frac{4 \cdot 4}{4 (1)}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.13.2.2

Cancel the common factor.

$\frac{85536}{5} + 18 (64 - \frac{4 \cdot 4}{4 \cdot 1}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.13.2.3

Rewrite the expression.

$\frac{85536}{5} + 18 (64 - \frac{4}{1}) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.13.2.4

Divide $4$ by $1$ .

$\frac{85536}{5} + 18 (64 - 1 \cdot 4) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.14

Multiply $- 1$ by $4$ .

$\frac{85536}{5} + 18 (64 - 4) + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.15

Subtract $4$ from $64$ .

$\frac{85536}{5} + 18 \cdot 60 + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.16

Multiply $18$ by $60$ .

$\frac{85536}{5} + 1080 + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.17

To write $1080$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{85536}{5} + 1080 \cdot \frac{5}{5} + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.18

Combine $1080$ and $\frac{5}{5}$ .

$\frac{85536}{5} + \frac{1080 \cdot 5}{5} + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.19

Combine the numerators over the common denominator.

$\frac{85536 + 1080 \cdot 5}{5} + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.20

Simplify the numerator.

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Step 3.10.4.20.1

Multiply $1080$ by $5$ .

$\frac{85536 + 5400}{5} + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.20.2

Add $85536$ and $5400$ .

$\frac{90936}{5} + (\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.21

Raise $4$ to the power of $3$ .

$\frac{90936}{5} + \frac{1}{3} \cdot 64 - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.22

Combine $\frac{1}{3}$ and $64$ .

$\frac{90936}{5} + \frac{64}{3} - \frac{1}{3} {(- 2)}^{3}$

Step 3.10.4.23

Raise $- 2$ to the power of $3$ .

$\frac{90936}{5} + \frac{64}{3} - \frac{1}{3} \cdot - 8$

Step 3.10.4.24

Multiply $- 8$ by $- 1$ .

$\frac{90936}{5} + \frac{64}{3} + 8 (\frac{1}{3})$

Step 3.10.4.25

Combine $8$ and $\frac{1}{3}$ .

$\frac{90936}{5} + \frac{64}{3} + \frac{8}{3}$

Step 3.10.4.26

Combine the numerators over the common denominator.

$\frac{90936}{5} + \frac{64 + 8}{3}$

Step 3.10.4.27

Add $64$ and $8$ .

$\frac{90936}{5} + \frac{72}{3}$

Step 3.10.4.28

Cancel the common factor of $72$ and $3$ .

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Step 3.10.4.28.1

Factor $3$ out of $72$ .

$\frac{90936}{5} + \frac{3 \cdot 24}{3}$

Step 3.10.4.28.2

Cancel the common factors.

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Step 3.10.4.28.2.1

Factor $3$ out of $3$ .

$\frac{90936}{5} + \frac{3 \cdot 24}{3 (1)}$

Step 3.10.4.28.2.2

Cancel the common factor.

$\frac{90936}{5} + \frac{3 \cdot 24}{3 \cdot 1}$

Step 3.10.4.28.2.3

Rewrite the expression.

$\frac{90936}{5} + \frac{24}{1}$

Step 3.10.4.28.2.4

Divide $24$ by $1$ .

$\frac{90936}{5} + 24$

Step 3.10.4.29

To write $24$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{90936}{5} + 24 \cdot \frac{5}{5}$

Step 3.10.4.30

Combine $24$ and $\frac{5}{5}$ .

$\frac{90936}{5} + \frac{24 \cdot 5}{5}$

Step 3.10.4.31

Combine the numerators over the common denominator.

$\frac{90936 + 24 \cdot 5}{5}$

Step 3.10.4.32

Simplify the numerator.

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Step 3.10.4.32.1

Multiply $24$ by $5$ .

$\frac{90936 + 120}{5}$

Step 3.10.4.32.2

Add $90936$ and $120$ .

$\frac{91056}{5}$

Step 4

Simplify the root mean square formula.

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Step 4.1

Multiply $\frac{1}{4 + 2}$ by $\frac{91056}{5}$ .

$f_{rms} = \sqrt{\frac{91056}{(4 + 2) \cdot 5}}$

Step 4.2

Add $4$ and $2$ .

$f_{rms} = \sqrt{\frac{91056}{6 \cdot 5}}$

Step 4.3

Reduce the expression $\frac{91056}{6 \cdot 5}$ by cancelling the common factors.

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Step 4.3.1

Factor $6$ out of $91056$ .

$f_{rms} = \sqrt{\frac{6 \cdot 15176}{6 \cdot 5}}$

Step 4.3.2

Factor $6$ out of $6 \cdot 5$ .

$f_{rms} = \sqrt{\frac{6 \cdot 15176}{6 (5)}}$

Step 4.3.3

Cancel the common factor.

$f_{rms} = \sqrt{\frac{6 \cdot 15176}{6 \cdot 5}}$

Step 4.3.4

Rewrite the expression.

$f_{rms} = \sqrt{\frac{15176}{5}}$

Step 4.4

Rewrite $\sqrt{\frac{15176}{5}}$ as $\frac{\sqrt{15176}}{\sqrt{5}}$ .

$f_{rms} = \frac{\sqrt{15176}}{\sqrt{5}}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Rewrite $15176$ as $2^{2} \cdot 3794$ .

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Step 4.5.1.1

Factor $4$ out of $15176$ .

$f_{rms} = \frac{\sqrt{4 (3794)}}{\sqrt{5}}$

Step 4.5.1.2

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

$f_{rms} = \frac{\sqrt{2^{2} \cdot 3794}}{\sqrt{5}}$

Step 4.5.2

Pull terms out from under the radical.

$f_{rms} = \frac{2 \sqrt{3794}}{\sqrt{5}}$

Step 4.6

Multiply $\frac{2 \sqrt{3794}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$f_{rms} = \frac{2 \sqrt{3794}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4.7

Combine and simplify the denominator.

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Step 4.7.1

Multiply $\frac{2 \sqrt{3794}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.7.2

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

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.7.3

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

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.7.4

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

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.7.5

Add $1$ and $1$ .

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.7.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.7.6.2

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

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.7.6.4

Cancel the common factor of $2$ .

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Step 4.7.6.4.1

Cancel the common factor.

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.7.6.4.2

Rewrite the expression.

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{5}$

Step 4.7.6.5

Evaluate the exponent.

$f_{rms} = \frac{2 \sqrt{3794} \sqrt{5}}{5}$

Step 4.8

Simplify the numerator.

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Step 4.8.1

Combine using the product rule for radicals.

$f_{rms} = \frac{2 \sqrt{5 \cdot 3794}}{5}$

Step 4.8.2

Multiply $5$ by $3794$ .

$f_{rms} = \frac{2 \sqrt{18970}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$f_{rms} = \frac{2 \sqrt{18970}}{5}$

Decimal Form:

$f_{rms} = 55.09264923 \dots$

Step 6