Basic Math Examples

$10^{- 3} = y \cdot 6.63 \cdot 10^{- 34} \cdot \frac{3 \cdot 10^{8}}{633 \cdot 10^{- 9}}$

Step 1

Rewrite the equation as $y \cdot 6.63 \cdot 10^{- 34} \cdot \frac{3 \cdot 10^{8}}{633 \cdot 10^{- 9}} = 10^{- 3}$ .

$y \cdot 6.63 \cdot 10^{- 34} \cdot \frac{3 \cdot 10^{8}}{633 \cdot 10^{- 9}} = 10^{- 3}$

Step 2

Simplify.

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Step 2.1

Move the decimal point in $633$ to the left by $2$ places and increase the power of $10^{- 9}$ by $2$ .

$y \cdot 6.63 \cdot 10^{- 34} \cdot \frac{3 \cdot 10^{8}}{6.33 \cdot 10^{- 7}} = 10^{- 3}$

Step 2.2

Reduce the expression $\frac{3 \cdot 10^{8}}{6.33 \cdot 10^{- 7}}$ by cancelling the common factors.

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Step 2.2.1

Factor $10^{- 7}$ out of $3 \cdot 10^{8}$ .

$y \cdot 6.63 \cdot 10^{- 34} \cdot \frac{10^{- 7} \cdot 3 \cdot 10^{15}}{6.33 \cdot 10^{- 7}} = 10^{- 3}$

Step 2.2.2

Factor $10^{- 7}$ out of $6.33 \cdot 10^{- 7}$ .

$y \cdot 6.63 \cdot 10^{- 34} \cdot \frac{10^{- 7} \cdot 3 \cdot 10^{15}}{10^{- 7} \cdot 6.33} = 10^{- 3}$

Step 2.2.3

Cancel the common factor.

$y \cdot 6.63 \cdot 10^{- 34} \cdot \frac{10^{- 7} \cdot 3 \cdot 10^{15}}{10^{- 7} \cdot 6.33} = 10^{- 3}$

Step 2.2.4

Rewrite the expression.

$y \cdot 6.63 \cdot 10^{- 34} \cdot \frac{3 \cdot 10^{15}}{6.33} = 10^{- 3}$

Step 2.3

Divide using scientific notation.

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Step 2.3.1

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

$y \cdot 6.63 \cdot 10^{- 34} \cdot ((\frac{3}{6.33}) (\frac{10^{15}}{1})) = 10^{- 3}$

Step 2.3.2

Divide $3$ by $6.33$ .

$y \cdot 6.63 \cdot 10^{- 34} \cdot (0.47393364 \frac{10^{15}}{1}) = 10^{- 3}$

Step 2.3.3

Divide $10^{15}$ by $1$ .

$y \cdot 6.63 \cdot 10^{- 34} \cdot 0.47393364 \cdot 10^{15} = 10^{- 3}$

Step 2.4

Move the decimal point in $0.47393364$ to the right by $1$ place and decrease the power of $10^{15}$ by $1$ .

$y \cdot 6.63 \cdot 10^{- 34} \cdot 4.73933649 \cdot 10^{14} = 10^{- 3}$

Step 2.5

Multiply $4.73933649$ by $6.63$ .

$y \cdot 31.42180094 \cdot 10^{- 34} \cdot 10^{14} = 10^{- 3}$

Step 2.6

Multiply $10^{- 34}$ by $10^{14}$ by adding the exponents.

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Step 2.6.1

Move $10^{14}$ .

$y \cdot (31.42180094 \cdot (10^{14} \cdot 10^{- 34})) = 10^{- 3}$

Step 2.6.2

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

$y \cdot (31.42180094 \cdot 10^{14 - 34}) = 10^{- 3}$

Step 2.6.3

Subtract $34$ from $14$ .

$y \cdot 31.42180094 \cdot 10^{- 20} = 10^{- 3}$

Step 3

Divide each term in $y \cdot 31.42180094 \cdot 10^{- 20} = 10^{- 3}$ by $3.14218009 \cdot 10^{- 19}$ and simplify.

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Step 3.1

Divide each term in $y \cdot 31.42180094 \cdot 10^{- 20} = 10^{- 3}$ by $3.14218009 \cdot 10^{- 19}$ .

$\frac{y \cdot 31.42180094 \cdot 10^{- 20}}{3.14218009 \cdot 10^{- 19}} = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Move $10^{- 20}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{y \cdot 31.42180094}{3.14218009 \cdot 10^{- 19} \cdot 10^{20}} = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

Multiply $10^{- 19}$ by $10^{20}$ by adding the exponents.

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Step 3.2.2.1.1

Move $10^{20}$ .

$\frac{y \cdot 31.42180094}{3.14218009 \cdot (10^{20} \cdot 10^{- 19})} = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.2.2.1.2

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

$\frac{y \cdot 31.42180094}{3.14218009 \cdot 10^{20 - 19}} = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.2.2.1.3

Subtract $19$ from $20$ .

$\frac{y \cdot 31.42180094}{3.14218009 \cdot 10^{1}} = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.2.2.2

Simplify $3.14218009 \cdot 10^{1}$ .

$\frac{y \cdot 31.42180094}{3.14218009 \cdot 10} = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.2.3

Multiply $3.14218009$ by $10$ .

$\frac{y \cdot 31.42180094}{31.42180094} = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.2.4

Cancel the common factor of $31.42180094$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{y \cdot 31.42180094}{31.42180094} = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.2.4.2

Divide $y$ by $1$ .

$y = \frac{10^{- 3}}{3.14218009 \cdot 10^{- 19}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide using scientific notation.

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Step 3.3.1.1

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

$y = (\frac{1}{3.14218009}) (\frac{10^{- 3}}{10^{- 19}})$

Step 3.3.1.2

Divide $1$ by $3.14218009$ .

$y = 0.31825037 \frac{10^{- 3}}{10^{- 19}}$

Step 3.3.1.3

Subtract the exponent from the denominator from the exponent of the numerator for the same base

$y = 0.31825037 \cdot 10^{- 3 - 19 \cdot - 1}$

Step 3.3.1.4

Multiply $- 19$ by $- 1$ .

$y = 0.31825037 \cdot 10^{- 3 + 19}$

Step 3.3.1.5

Add $- 3$ and $19$ .

$y = 0.31825037 \cdot 10^{16}$

Step 3.3.2

Move the decimal point in $0.31825037$ to the right by $1$ place and decrease the power of $10^{16}$ by $1$ .

$y = 3.18250377 \cdot 10^{15}$