Basic Math Examples

$- 0.0314 {ms}^{- 1} = \frac{\sqrt{F}}{\sqrt{1.3 (10^{- 4}) kg m^{- 1}}}$

Step 1

Rewrite the equation as $\frac{\sqrt{F}}{\sqrt{1.3 (10^{- 4}) kg m^{- 1}}} = - 0.0314 {ms}^{- 1}$ .

$\frac{\sqrt{F}}{\sqrt{1.3 (10^{- 4}) kg m^{- 1}}} = - 0.0314 {ms}^{- 1}$

Step 2

Cross multiply.

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Step 2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$- 0.0314 {ms}^{- 1} \cdot \sqrt{1.3 (10^{- 4}) kg m^{- 1}} = \sqrt{F}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 0.0314 {ms}^{- 1} \cdot \sqrt{1.3 (10^{- 4}) kg m^{- 1}}$ .

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Step 2.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 0.0314 {ms}^{- 1} \cdot \sqrt{1.3 (10^{- 4}) kg \frac{1}{m}} = \sqrt{F}$

Step 2.2.1.2

Combine $1.3 (10^{- 4}) kg$ and $\frac{1}{m}$ .

$- 0.0314 {ms}^{- 1} \cdot \sqrt{\frac{1.3 (10^{- 4}) kg}{m}} = \sqrt{F}$

Step 2.2.1.3

Rewrite $\sqrt{\frac{1.3 (10^{- 4}) kg}{m}}$ as $\frac{\sqrt{1.3 (10^{- 4}) kg}}{\sqrt{m}}$ .

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg}}{\sqrt{m}} = \sqrt{F}$

Step 2.2.1.4

Multiply $\frac{\sqrt{1.3 (10^{- 4}) kg}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$- 0.0314 {ms}^{- 1} \cdot (\frac{\sqrt{1.3 (10^{- 4}) kg}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}) = \sqrt{F}$

Step 2.2.1.5

Combine and simplify the denominator.

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Step 2.2.1.5.1

Multiply $\frac{\sqrt{1.3 (10^{- 4}) kg}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{\sqrt{m} \sqrt{m}} = \sqrt{F}$

Step 2.2.1.5.2

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

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}} = \sqrt{F}$

Step 2.2.1.5.3

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

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}} = \sqrt{F}$

Step 2.2.1.5.4

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

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{{\sqrt{m}}^{1 + 1}} = \sqrt{F}$

Step 2.2.1.5.5

Add $1$ and $1$ .

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{{\sqrt{m}}^{2}} = \sqrt{F}$

Step 2.2.1.5.6

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

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Step 2.2.1.5.6.1

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

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}} = \sqrt{F}$

Step 2.2.1.5.6.2

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

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}} = \sqrt{F}$

Step 2.2.1.5.6.3

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

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{m^{\frac{2}{2}}} = \sqrt{F}$

Step 2.2.1.5.6.4

Cancel the common factor of $2$ .

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Step 2.2.1.5.6.4.1

Cancel the common factor.

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{m^{\frac{2}{2}}} = \sqrt{F}$

Step 2.2.1.5.6.4.2

Rewrite the expression.

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{m^{1}} = \sqrt{F}$

Step 2.2.1.5.6.5

Simplify.

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg} \sqrt{m}}{m} = \sqrt{F}$

Step 2.2.1.6

Combine using the product rule for radicals.

$- 0.0314 {ms}^{- 1} \cdot \frac{\sqrt{1.3 (10^{- 4}) kg m}}{m} = \sqrt{F}$

Step 2.2.1.7

Combine $- 0.0314 {ms}^{- 1}$ and $\frac{\sqrt{1.3 (10^{- 4}) kg m}}{m}$ .

$\frac{- 0.0314 {ms}^{- 1} \sqrt{1.3 (10^{- 4}) kg m}}{m} = \sqrt{F}$

Step 3

Solve for $- 0.0314 {ms}^{- 1} \sqrt{1.3 (10^{- 4}) kg m}$ .

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Step 3.1

Multiply both sides by $m$ .

$\frac{- 0.0314 {ms}^{- 1} \sqrt{1.3 (10^{- 4}) kg m}}{m} m = \sqrt{F} m$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $m$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 0.0314 {ms}^{- 1} \sqrt{1.3 (10^{- 4}) kg m}}{m} m = \sqrt{F} m$

Step 3.2.1.2

Rewrite the expression.

$- 0.0314 {ms}^{- 1} \sqrt{1.3 (10^{- 4}) kg m} = \sqrt{F} m$

Step 4

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 0.0314 {ms}^{- 1} \sqrt{1.3 (10^{- 4}) kg m})}^{2} = {(\sqrt{F} m)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1.3 (10^{- 4}) kg m}$ as ${(1.3 (10^{- 4}) kg m)}^{\frac{1}{2}}$ .

${(- 0.0314 {ms}^{- 1} {(1.3 (10^{- 4}) kg m)}^{\frac{1}{2}})}^{2} = {(\sqrt{F} m)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(- 0.0314 {ms}^{- 1} {(1.3 (10^{- 4}) kg m)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $1.3 (10^{- 4}) kg m$ .

${(- 0.0314 {ms}^{- 1} ({(1.3 (10^{- 4}) kg)}^{\frac{1}{2}} m^{\frac{1}{2}}))}^{2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.2.1

Apply the product rule to $- 0.0314 {ms}^{- 1} {(1.3 (10^{- 4}) kg)}^{\frac{1}{2}} m^{\frac{1}{2}}$ .

${(- 0.0314 {ms}^{- 1} {(1.3 (10^{- 4}) kg)}^{\frac{1}{2}})}^{2} {(m^{\frac{1}{2}})}^{2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.2.2

Apply the product rule to $- 0.0314 {ms}^{- 1} {(1.3 (10^{- 4}) kg)}^{\frac{1}{2}}$ .

${(- 0.0314 {ms}^{- 1})}^{2} {({(1.3 (10^{- 4}) kg)}^{\frac{1}{2}})}^{2} {(m^{\frac{1}{2}})}^{2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.3

Multiply the exponents in ${({(1.3 (10^{- 4}) kg)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.3.1

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

${(- 0.0314 {ms}^{- 1})}^{2} {(1.3 (10^{- 4}) kg)}^{\frac{1}{2} \cdot 2} {(m^{\frac{1}{2}})}^{2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

${(- 0.0314 {ms}^{- 1})}^{2} {(1.3 (10^{- 4}) kg)}^{\frac{1}{2} \cdot 2} {(m^{\frac{1}{2}})}^{2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

${(- 0.0314 {ms}^{- 1})}^{2} {(1.3 (10^{- 4}) kg)}^{1} {(m^{\frac{1}{2}})}^{2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.4

Simplify.

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) {(m^{\frac{1}{2}})}^{2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.5

Multiply the exponents in ${(m^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.5.1

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

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m^{\frac{1}{2} \cdot 2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.5.2

Cancel the common factor of $2$ .

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Step 5.2.1.5.2.1

Cancel the common factor.

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m^{\frac{1}{2} \cdot 2} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.5.2.2

Rewrite the expression.

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m^{1} = {(\sqrt{F} m)}^{2}$

Step 5.2.1.6

Simplify.

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m = {(\sqrt{F} m)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(\sqrt{F} m)}^{2}$ .

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Step 5.3.1.1

Apply the product rule to $\sqrt{F} m$ .

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m = {\sqrt{F}}^{2} m^{2}$

Step 5.3.1.2

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

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Step 5.3.1.2.1

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

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m = {(F^{\frac{1}{2}})}^{2} m^{2}$

Step 5.3.1.2.2

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

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m = F^{\frac{1}{2} \cdot 2} m^{2}$

Step 5.3.1.2.3

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

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m = F^{\frac{2}{2}} m^{2}$

Step 5.3.1.2.4

Cancel the common factor of $2$ .

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Step 5.3.1.2.4.1

Cancel the common factor.

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m = F^{\frac{2}{2}} m^{2}$

Step 5.3.1.2.4.2

Rewrite the expression.

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m = F^{1} m^{2}$

Step 5.3.1.2.5

Simplify.

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m = F m^{2}$

Step 6

Solve for $m$ .

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Step 6.1

Subtract $F m^{2}$ from both sides of the equation.

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m - F m^{2} = 0$

Step 6.2

Factor $m$ out of ${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m - F m^{2}$ .

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Step 6.2.1

Factor $m$ out of ${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) m$ .

$m ({(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)) - F m^{2} = 0$

Step 6.2.2

Factor $m$ out of $- F m^{2}$ .

$m ({(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)) + m (- F m) = 0$

Step 6.2.3

Factor $m$ out of $m ({(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)) + m (- F m)$ .

$m ({(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) - F m) = 0$

Step 6.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$m = 0$

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) - F m = 0$

Step 6.4

Set $m$ equal to $0$ .

$m = 0$

Step 6.5

Set ${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) - F m$ equal to $0$ and solve for $m$ .

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Step 6.5.1

Set ${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) - F m$ equal to $0$ .

${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) - F m = 0$

Step 6.5.2

Solve ${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) - F m = 0$ for $m$ .

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Step 6.5.2.1

Subtract ${(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)$ from both sides of the equation.

$- F m = - {(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)$

Step 6.5.2.2

Divide each term in $- F m = - {(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)$ by $- F$ and simplify.

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Step 6.5.2.2.1

Divide each term in $- F m = - {(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)$ by $- F$ .

$\frac{- F m}{- F} = \frac{- {(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)}{- F}$

Step 6.5.2.2.2

Simplify the left side.

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Step 6.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{F m}{F} = \frac{- {(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)}{- F}$

Step 6.5.2.2.2.2

Cancel the common factor of $F$ .

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Step 6.5.2.2.2.2.1

Cancel the common factor.

$\frac{F m}{F} = \frac{- {(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)}{- F}$

Step 6.5.2.2.2.2.2

Divide $m$ by $1$ .

$m = \frac{- {(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)}{- F}$

Step 6.5.2.2.3

Simplify the right side.

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Step 6.5.2.2.3.1

Dividing two negative values results in a positive value.

$m = \frac{{(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)}{F}$

Step 6.6

The final solution is all the values that make $m ({(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg) - F m) = 0$ true.

$m = 0, \frac{{(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)}{F}$

Step 7

Simplify $m = 0, \frac{{(- 0.0314 {ms}^{- 1})}^{2} (1.3 (10^{- 4}) kg)}{F}$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the product rule to $- 0.0314 {ms}^{- 1}$ .

$m = 0, \frac{{(- 0.0314)}^{2} {({ms}^{- 1})}^{2} \cdot 1.3 \cdot 10^{- 4} kg}{F}$

Step 7.1.2

Multiply $1.3$ by ${(- 0.0314)}^{2}$ .

$m = 0, \frac{0.00128174 {({ms}^{- 1})}^{2} \cdot 10^{- 4} kg}{F}$

Step 7.1.3

Multiply the exponents in ${({ms}^{- 1})}^{2}$ .

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Step 7.1.3.1

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

$m = 0, \frac{0.00128174 {ms}^{- 1 \cdot 2} \cdot 10^{- 4} kg}{F}$

Step 7.1.3.2

Multiply $- 1$ by $2$ .

$m = 0, \frac{0.00128174 {ms}^{- 2} \cdot 10^{- 4} kg}{F}$

Step 7.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$m = 0, \frac{0.00128174 \frac{1}{{ms}^{2}} \cdot 10^{- 4} kg}{F}$

Step 7.1.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$m = 0, \frac{0.00128174 \frac{1}{{ms}^{2}} \cdot \frac{1}{10^{4}} kg}{F}$

Step 7.1.6

Raise $10$ to the power of $4$ .

$m = 0, \frac{0.00128174 \frac{1}{{ms}^{2}} \cdot \frac{1}{10000} kg}{F}$

Step 7.1.7

Combine exponents.

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Step 7.1.7.1

Combine $\frac{1}{10000}$ and $0.00128174$ .

$m = 0, \frac{\frac{0.00128174}{10000} \frac{1}{{ms}^{2}} kg}{F}$

Step 7.1.7.2

Combine $kg$ and $\frac{1}{{ms}^{2}}$ .

$m = 0, \frac{\frac{0.00128174}{10000} \frac{kg}{{ms}^{2}}}{F}$

Step 7.1.8

Divide $0.00128174$ by $10000$ .

$m = 0, \frac{0.00000012 \frac{kg}{{ms}^{2}}}{F}$

Step 7.2

Factor $\frac{kg}{{ms}^{2}}$ out of $\frac{0.00000012 \frac{kg}{{ms}^{2}}}{F}$ .

$m = 0, \frac{kg}{{ms}^{2}} \cdot \frac{0.00000012}{F}$

Step 7.3

Combine fractions.

$m = 0, \frac{0.00000012}{F} \frac{kg}{{ms}^{2}}$