Calculus Examples

$y = x^{2} - 8 x$ , $[0, 5]$

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.

$f_{rms} = \sqrt{\frac{1}{5 - 0} \cdot (\int_{0}^{5} {(x^{2} - 8 x)}^{2} d x)}$

Step 3

Evaluate the integral.

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Step 3.1

Expand ${(x^{2} - 8 x)}^{2}$ .

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Step 3.1.1

Rewrite ${(x^{2} - 8 x)}^{2}$ as $(x^{2} - 8 x) (x^{2} - 8 x)$ .

$\int_{0}^{5} (x^{2} - 8 x) (x^{2} - 8 x) d x$

Step 3.1.2

Apply the distributive property.

$\int_{0}^{5} x^{2} (x^{2} - 8 x) - 8 x (x^{2} - 8 x) d x$

Step 3.1.3

Apply the distributive property.

$\int_{0}^{5} x^{2} x^{2} + x^{2} (- 8 x) - 8 x (x^{2} - 8 x) d x$

Step 3.1.4

Apply the distributive property.

$\int_{0}^{5} x^{2} x^{2} + x^{2} (- 8 x) - 8 x \cdot x^{2} - 8 x (- 8 x) d x$

Step 3.1.5

Reorder $x^{2}$ and $- 8$ .

$\int_{0}^{5} x^{2} x^{2} - 8 \cdot x^{2} x - 8 x \cdot x^{2} - 8 x (- 8 x) d x$

Step 3.1.6

Move $x$ .

$\int_{0}^{5} x^{2} x^{2} - 8 \cdot x^{2} x - 8 x \cdot x^{2} - 8 \cdot - 8 x \cdot x d x$

Step 3.1.7

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

$\int_{0}^{5} x^{2 + 2} - 8 x^{2} x - 8 x \cdot x^{2} - 8 \cdot - 8 x \cdot x d x$

Step 3.1.8

Add $2$ and $2$ .

$\int_{0}^{5} x^{4} - 8 x^{2} x - 8 x \cdot x^{2} - 8 \cdot - 8 x \cdot x d x$

Step 3.1.9

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

$\int_{0}^{5} x^{4} - 8 (x^{2} x^{1}) - 8 x \cdot x^{2} - 8 \cdot - 8 x \cdot x d x$

Step 3.1.10

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

$\int_{0}^{5} x^{4} - 8 x^{2 + 1} - 8 x \cdot x^{2} - 8 \cdot - 8 x \cdot x d x$

Step 3.1.11

Add $2$ and $1$ .

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 x \cdot x^{2} - 8 \cdot - 8 x \cdot x d x$

Step 3.1.12

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

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 (x^{1} x^{2}) - 8 \cdot - 8 x \cdot x d x$

Step 3.1.13

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

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 x^{1 + 2} - 8 \cdot - 8 x \cdot x d x$

Step 3.1.14

Add $1$ and $2$ .

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 x^{3} - 8 \cdot - 8 x \cdot x d x$

Step 3.1.15

Multiply $- 8$ by $- 8$ .

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 x^{3} + 64 x \cdot x d x$

Step 3.1.16

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

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 x^{3} + 64 (x^{1} x) d x$

Step 3.1.17

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

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 x^{3} + 64 (x^{1} x^{1}) d x$

Step 3.1.18

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

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 x^{3} + 64 x^{1 + 1} d x$

Step 3.1.19

Add $1$ and $1$ .

$\int_{0}^{5} x^{4} - 8 x^{3} - 8 x^{3} + 64 x^{2} d x$

Step 3.1.20

Subtract $8 x^{3}$ from $- 8 x^{3}$ .

$\int_{0}^{5} x^{4} - 16 x^{3} + 64 x^{2} d x$

Step 3.2

Split the single integral into multiple integrals.

$\int_{0}^{5} x^{4} d x + \int_{0}^{5} - 16 x^{3} d x + \int_{0}^{5} 64 x^{2} d x$

Step 3.3

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

$\frac{1}{5} x^{5}]_{0}^{5} + \int_{0}^{5} - 16 x^{3} d x + \int_{0}^{5} 64 x^{2} d x$

Step 3.4

Since $- 16$ is constant with respect to $x$ , move $- 16$ out of the integral.

$\frac{1}{5} x^{5}]_{0}^{5} - 16 \int_{0}^{5} x^{3} d x + \int_{0}^{5} 64 x^{2} d x$

Step 3.5

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

$\frac{1}{5} x^{5}]_{0}^{5} - 16 (\frac{1}{4} x^{4}]_{0}^{5}) + \int_{0}^{5} 64 x^{2} d x$

Step 3.6

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

$\frac{1}{5} x^{5}]_{0}^{5} - 16 (\frac{x^{4}}{4}]_{0}^{5}) + \int_{0}^{5} 64 x^{2} d x$

Step 3.7

Since $64$ is constant with respect to $x$ , move $64$ out of the integral.

$\frac{1}{5} x^{5}]_{0}^{5} - 16 (\frac{x^{4}}{4}]_{0}^{5}) + 64 \int_{0}^{5} x^{2} d x$

Step 3.8

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

$\frac{1}{5} x^{5}]_{0}^{5} - 16 (\frac{x^{4}}{4}]_{0}^{5}) + 64 (\frac{1}{3} x^{3}]_{0}^{5})$

Step 3.9

Simplify the answer.

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Step 3.9.1

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

$\frac{1}{5} x^{5}]_{0}^{5} - 16 (\frac{x^{4}}{4}]_{0}^{5}) + 64 (\frac{x^{3}}{3}]_{0}^{5})$

Step 3.9.2

Substitute and simplify.

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Step 3.9.2.1

Evaluate $\frac{1}{5} x^{5}$ at $5$ and at $0$ .

$(\frac{1}{5} \cdot 5^{5}) - \frac{1}{5} \cdot 0^{5} - 16 (\frac{x^{4}}{4}]_{0}^{5}) + 64 (\frac{x^{3}}{3}]_{0}^{5})$

Step 3.9.2.2

Evaluate $\frac{x^{4}}{4}$ at $5$ and at $0$ .

$\frac{1}{5} \cdot 5^{5} - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 (\frac{x^{3}}{3}]_{0}^{5})$

Step 3.9.2.3

Evaluate $\frac{x^{3}}{3}$ at $5$ and at $0$ .

$\frac{1}{5} \cdot 5^{5} - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4

Simplify.

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Step 3.9.2.4.1

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

$\frac{1}{5} \cdot 3125 - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.2

Combine $\frac{1}{5}$ and $3125$ .

$\frac{3125}{5} - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.3

Cancel the common factor of $3125$ and $5$ .

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Step 3.9.2.4.3.1

Factor $5$ out of $3125$ .

$\frac{5 \cdot 625}{5} - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.3.2

Cancel the common factors.

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Step 3.9.2.4.3.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 625}{5 (1)} - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.3.2.2

Cancel the common factor.

$\frac{5 \cdot 625}{5 \cdot 1} - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.3.2.3

Rewrite the expression.

$\frac{625}{1} - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.3.2.4

Divide $625$ by $1$ .

$625 - \frac{1}{5} \cdot 0^{5} - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.4

Raising $0$ to any positive power yields $0$ .

$625 - \frac{1}{5} \cdot 0 - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.5

Multiply $0$ by $- 1$ .

$625 + 0 (\frac{1}{5}) - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.6

Multiply $0$ by $\frac{1}{5}$ .

$625 + 0 - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.7

Add $625$ and $0$ .

$625 - 16 (\frac{5^{4}}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.8

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

$625 - 16 (\frac{625}{4} - \frac{0^{4}}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.9

Raising $0$ to any positive power yields $0$ .

$625 - 16 (\frac{625}{4} - \frac{0}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.10

Cancel the common factor of $0$ and $4$ .

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Step 3.9.2.4.10.1

Factor $4$ out of $0$ .

$625 - 16 (\frac{625}{4} - \frac{4 (0)}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.10.2

Cancel the common factors.

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Step 3.9.2.4.10.2.1

Factor $4$ out of $4$ .

$625 - 16 (\frac{625}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.10.2.2

Cancel the common factor.

$625 - 16 (\frac{625}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.10.2.3

Rewrite the expression.

$625 - 16 (\frac{625}{4} - \frac{0}{1}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.10.2.4

Divide $0$ by $1$ .

$625 - 16 (\frac{625}{4} - 0) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.11

Multiply $- 1$ by $0$ .

$625 - 16 (\frac{625}{4} + 0) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.12

Add $\frac{625}{4}$ and $0$ .

$625 - 16 (\frac{625}{4}) + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.13

Combine $- 16$ and $\frac{625}{4}$ .

$625 + \frac{- 16 \cdot 625}{4} + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.14

Multiply $- 16$ by $625$ .

$625 + \frac{- 10000}{4} + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.15

Cancel the common factor of $- 10000$ and $4$ .

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Step 3.9.2.4.15.1

Factor $4$ out of $- 10000$ .

$625 + \frac{4 \cdot - 2500}{4} + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.15.2

Cancel the common factors.

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Step 3.9.2.4.15.2.1

Factor $4$ out of $4$ .

$625 + \frac{4 \cdot - 2500}{4 (1)} + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.15.2.2

Cancel the common factor.

$625 + \frac{4 \cdot - 2500}{4 \cdot 1} + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.15.2.3

Rewrite the expression.

$625 + \frac{- 2500}{1} + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.15.2.4

Divide $- 2500$ by $1$ .

$625 - 2500 + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.16

Subtract $2500$ from $625$ .

$- 1875 + 64 ((\frac{5^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.9.2.4.17

Raise $5$ to the power of $3$ .

$- 1875 + 64 (\frac{125}{3} - \frac{0^{3}}{3})$

Step 3.9.2.4.18

Raising $0$ to any positive power yields $0$ .

$- 1875 + 64 (\frac{125}{3} - \frac{0}{3})$

Step 3.9.2.4.19

Cancel the common factor of $0$ and $3$ .

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Step 3.9.2.4.19.1

Factor $3$ out of $0$ .

$- 1875 + 64 (\frac{125}{3} - \frac{3 (0)}{3})$

Step 3.9.2.4.19.2

Cancel the common factors.

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Step 3.9.2.4.19.2.1

Factor $3$ out of $3$ .

$- 1875 + 64 (\frac{125}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.9.2.4.19.2.2

Cancel the common factor.

$- 1875 + 64 (\frac{125}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.9.2.4.19.2.3

Rewrite the expression.

$- 1875 + 64 (\frac{125}{3} - \frac{0}{1})$

Step 3.9.2.4.19.2.4

Divide $0$ by $1$ .

$- 1875 + 64 (\frac{125}{3} - 0)$

Step 3.9.2.4.20

Multiply $- 1$ by $0$ .

$- 1875 + 64 (\frac{125}{3} + 0)$

Step 3.9.2.4.21

Add $\frac{125}{3}$ and $0$ .

$- 1875 + 64 (\frac{125}{3})$

Step 3.9.2.4.22

Combine $64$ and $\frac{125}{3}$ .

$- 1875 + \frac{64 \cdot 125}{3}$

Step 3.9.2.4.23

Multiply $64$ by $125$ .

$- 1875 + \frac{8000}{3}$

Step 3.9.2.4.24

To write $- 1875$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 1875 \cdot \frac{3}{3} + \frac{8000}{3}$

Step 3.9.2.4.25

Combine $- 1875$ and $\frac{3}{3}$ .

$\frac{- 1875 \cdot 3}{3} + \frac{8000}{3}$

Step 3.9.2.4.26

Combine the numerators over the common denominator.

$\frac{- 1875 \cdot 3 + 8000}{3}$

Step 3.9.2.4.27

Simplify the numerator.

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Step 3.9.2.4.27.1

Multiply $- 1875$ by $3$ .

$\frac{- 5625 + 8000}{3}$

Step 3.9.2.4.27.2

Add $- 5625$ and $8000$ .

$\frac{2375}{3}$

Step 4

Simplify the root mean square formula.

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Step 4.1

Multiply $\frac{1}{5 - 0}$ by $\frac{2375}{3}$ .

$f_{rms} = \sqrt{\frac{2375}{(5 - 0) \cdot 3}}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Multiply $- 1$ by $0$ .

$f_{rms} = \sqrt{\frac{2375}{(5 + 0) \cdot 3}}$

Step 4.2.2

Add $5$ and $0$ .

$f_{rms} = \sqrt{\frac{2375}{5 \cdot 3}}$

Step 4.3

Reduce the expression $\frac{2375}{5 \cdot 3}$ by cancelling the common factors.

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Step 4.3.1

Factor $5$ out of $2375$ .

$f_{rms} = \sqrt{\frac{5 \cdot 475}{5 \cdot 3}}$

Step 4.3.2

Factor $5$ out of $5 \cdot 3$ .

$f_{rms} = \sqrt{\frac{5 \cdot 475}{5 (3)}}$

Step 4.3.3

Cancel the common factor.

$f_{rms} = \sqrt{\frac{5 \cdot 475}{5 \cdot 3}}$

Step 4.3.4

Rewrite the expression.

$f_{rms} = \sqrt{\frac{475}{3}}$

Step 4.4

Rewrite $\sqrt{\frac{475}{3}}$ as $\frac{\sqrt{475}}{\sqrt{3}}$ .

$f_{rms} = \frac{\sqrt{475}}{\sqrt{3}}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Rewrite $475$ as $5^{2} \cdot 19$ .

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Step 4.5.1.1

Factor $25$ out of $475$ .

$f_{rms} = \frac{\sqrt{25 (19)}}{\sqrt{3}}$

Step 4.5.1.2

Rewrite $25$ as $5^{2}$ .

$f_{rms} = \frac{\sqrt{5^{2} \cdot 19}}{\sqrt{3}}$

Step 4.5.2

Pull terms out from under the radical.

$f_{rms} = \frac{5 \sqrt{19}}{\sqrt{3}}$

Step 4.6

Multiply $\frac{5 \sqrt{19}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f_{rms} = \frac{5 \sqrt{19}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.7

Combine and simplify the denominator.

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Step 4.7.1

Multiply $\frac{5 \sqrt{19}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.7.2

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

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.7.3

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

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.7.4

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

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.7.5

Add $1$ and $1$ .

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.7.6

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

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Step 4.7.6.1

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

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.7.6.2

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

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.7.6.3

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

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{3^{\frac{2}{2}}}$

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.

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.7.6.4.2

Rewrite the expression.

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{3}$

Step 4.7.6.5

Evaluate the exponent.

$f_{rms} = \frac{5 \sqrt{19} \sqrt{3}}{3}$

Step 4.8

Simplify the numerator.

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Step 4.8.1

Combine using the product rule for radicals.

$f_{rms} = \frac{5 \sqrt{3 \cdot 19}}{3}$

Step 4.8.2

Multiply $3$ by $19$ .

$f_{rms} = \frac{5 \sqrt{57}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$f_{rms} = \frac{5 \sqrt{57}}{3}$

Decimal Form:

$f_{rms} = 12.58305739 \dots$

Step 6