Calculus Examples

$\frac{n_{1}}{2 \cdot 0.2} \cdot \sqrt{\frac{30}{3 \cdot 10^{- 3}}} = \frac{n_{2}}{2 \cdot 0.5} \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1

Simplify $\frac{n_{1}}{2 \cdot 0.2} \cdot \sqrt{\frac{30}{3 \cdot 10^{- 3}}}$ .

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Step 1.1

Divide using scientific notation.

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Step 1.1.1

Group coefficients together and exponents together to divide numbers in scientific notation.

$\frac{n_{1}}{2 \cdot 0.2} \cdot \sqrt{(\frac{30}{3}) (\frac{1}{10^{- 3}})} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.1.2

Divide $30$ by $3$ .

$\frac{n_{1}}{2 \cdot 0.2} \cdot \sqrt{10 \frac{1}{10^{- 3}}} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.1.3

Move $10^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{n_{1}}{2 \cdot 0.2} \cdot \sqrt{10 \cdot 10^{3}} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.2

Rewrite $10 \cdot 10^{3}$ as $1 \cdot 10^{4}$ .

$\frac{n_{1}}{2 \cdot 0.2} \cdot \sqrt{1 \cdot 10^{4}} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.3

Rewrite $\sqrt{1 \cdot 10^{4}}$ as $\sqrt{1} \cdot \sqrt{10^{4}}$ .

$\frac{n_{1}}{2 \cdot 0.2} \cdot (\sqrt{1} \cdot \sqrt{10^{4}}) = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.4

Evaluate the root.

$\frac{n_{1}}{2 \cdot 0.2} \cdot (1 \cdot \sqrt{10^{4}}) = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.5

Rewrite $10^{4}$ as ${(10^{2})}^{2}$ .

$\frac{n_{1}}{2 \cdot 0.2} \cdot (1 \cdot \sqrt{{(10^{2})}^{2}}) = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{n_{1}}{2 \cdot 0.2} \cdot 1 \cdot 10^{2} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.7

Multiply $2$ by $0.2$ .

$\frac{n_{1}}{0.4} \cdot 1 \cdot 10^{2} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.8

Multiply by $1$ .

$\frac{1 n_{1}}{0.4} \cdot 1 \cdot 10^{2} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.9

Factor $0.4$ out of $0.4$ .

$\frac{1 n_{1}}{0.4 (1)} \cdot 1 \cdot 10^{2} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.10

Separate fractions.

$\frac{1}{0.4} \cdot \frac{n_{1}}{1} \cdot 1 \cdot 10^{2} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.11

Divide $1$ by $0.4$ .

$2.5 \frac{n_{1}}{1} \cdot 1 \cdot 10^{2} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.12

Divide $n_{1}$ by $1$ .

$2.5 n_{1} \cdot 1 \cdot 10^{2} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.13

Multiply $10^{2}$ by $1$ .

$2.5 n_{1} \cdot 10^{2} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.14

Raise $10$ to the power of $2$ .

$2.5 n_{1} \cdot 100 = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 1.15

Multiply $100$ by $2.5$ .

$250 n_{1} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$

Step 2

Simplify $\frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{\frac{30}{2.7 \cdot 10^{- 4}}}$ .

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Step 2.1

Divide using scientific notation.

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Step 2.1.1

Group coefficients together and exponents together to divide numbers in scientific notation.

$250 n_{1} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{(\frac{30}{2.7}) (\frac{1}{10^{- 4}})}$

Step 2.1.2

Divide $30$ by $2.7$ .

$250 n_{1} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{11.\overline{1} \frac{1}{10^{- 4}}}$

Step 2.1.3

Move $10^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$250 n_{1} = \frac{n_{2}}{2 \cdot 0.5} \cdot \sqrt{11.\overline{1} \cdot 10^{4}}$

Step 2.2

Rewrite $\sqrt{11.\overline{1} \cdot 10^{4}}$ as $\sqrt{11.\overline{1}} \cdot \sqrt{10^{4}}$ .

$250 n_{1} = \frac{n_{2}}{2 \cdot 0.5} \cdot (\sqrt{11.\overline{1}} \cdot \sqrt{10^{4}})$

Step 2.3

Evaluate the root.

$250 n_{1} = \frac{n_{2}}{2 \cdot 0.5} \cdot (3.\overline{3} \cdot \sqrt{10^{4}})$

Step 2.4

Rewrite $10^{4}$ as ${(10^{2})}^{2}$ .

$250 n_{1} = \frac{n_{2}}{2 \cdot 0.5} \cdot (3.\overline{3} \cdot \sqrt{{(10^{2})}^{2}})$

Step 2.5

Pull terms out from under the radical, assuming positive real numbers.

$250 n_{1} = \frac{n_{2}}{2 \cdot 0.5} \cdot 3.\overline{3} \cdot 10^{2}$

Step 2.6

Multiply $2$ by $0.5$ .

$250 n_{1} = \frac{n_{2}}{1} \cdot 3.\overline{3} \cdot 10^{2}$

Step 2.7

Divide $n_{2}$ by $1$ .

$250 n_{1} = n_{2} \cdot 3.\overline{3} \cdot 10^{2}$

Step 2.8

Move $3.\overline{3}$ to the left of $n_{2}$ .

$250 n_{1} = 3.\overline{3} \cdot n_{2} \cdot 10^{2}$

Step 2.9

Raise $10$ to the power of $2$ .

$250 n_{1} = 3.\overline{3} n_{2} \cdot 100$

Step 2.10

Multiply $100$ by $3.\overline{3}$ .

$250 n_{1} = 333.\overline{3} n_{2}$

Step 3

Divide each term in $250 n_{1} = 333.\overline{3} n_{2}$ by $250$ and simplify.

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Step 3.1

Divide each term in $250 n_{1} = 333.\overline{3} n_{2}$ by $250$ .

$\frac{250 n_{1}}{250} = \frac{333.\overline{3} n_{2}}{250}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $250$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{250 n_{1}}{250} = \frac{333.\overline{3} n_{2}}{250}$

Step 3.2.1.2

Divide $n_{1}$ by $1$ .

$n_{1} = \frac{333.\overline{3} n_{2}}{250}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Factor $333.\overline{3}$ out of $333.\overline{3} n_{2}$ .

$n_{1} = \frac{333.\overline{3} n_{2}}{250}$

Step 3.3.2

Factor $250$ out of $250$ .

$n_{1} = \frac{333.\overline{3} n_{2}}{250 (1)}$

Step 3.3.3

Separate fractions.

$n_{1} = \frac{333.\overline{3}}{250} \cdot \frac{n_{2}}{1}$

Step 3.3.4

Divide $333.\overline{3}$ by $250$ .

$n_{1} = 1.\overline{3} \frac{n_{2}}{1}$

Step 3.3.5

Divide $n_{2}$ by $1$ .

$n_{1} = 1.\overline{3} n_{2}$