Basic Math Examples

$0.2 \cdot \frac{0.1}{\sqrt{c}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1

Simplify both sides.

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Step 1.1

Multiply $\frac{0.1}{\sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$0.2 \cdot (\frac{0.1}{\sqrt{c}} \cdot \frac{\sqrt{c}}{\sqrt{c}}) \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{0.1}{\sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$0.2 \cdot \frac{0.1 \sqrt{c}}{\sqrt{c} \sqrt{c}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.2

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

$0.2 \cdot \frac{0.1 \sqrt{c}}{{\sqrt{c}}^{1} \sqrt{c}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.3

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

$0.2 \cdot \frac{0.1 \sqrt{c}}{{\sqrt{c}}^{1} {\sqrt{c}}^{1}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.4

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

$0.2 \cdot \frac{0.1 \sqrt{c}}{{\sqrt{c}}^{1 + 1}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.5

Add $1$ and $1$ .

$0.2 \cdot \frac{0.1 \sqrt{c}}{{\sqrt{c}}^{2}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.6

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

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Step 1.2.6.1

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

$0.2 \cdot \frac{0.1 \sqrt{c}}{{(c^{\frac{1}{2}})}^{2}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.6.2

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

$0.2 \cdot \frac{0.1 \sqrt{c}}{c^{\frac{1}{2} \cdot 2}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.6.3

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

$0.2 \cdot \frac{0.1 \sqrt{c}}{c^{\frac{2}{2}}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$0.2 \cdot \frac{0.1 \sqrt{c}}{c^{\frac{2}{2}}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.6.4.2

Rewrite the expression.

$0.2 \cdot \frac{0.1 \sqrt{c}}{c^{1}} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.2.6.5

Simplify.

$0.2 \cdot \frac{0.1 \sqrt{c}}{c} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.3

Multiply $0.2 \frac{0.1 \sqrt{c}}{c}$ .

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Step 1.3.1

Combine $0.2$ and $\frac{0.1 \sqrt{c}}{c}$ .

$\frac{0.2 (0.1 \sqrt{c})}{c} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.3.2

Multiply $0.1$ by $0.2$ .

$\frac{0.02 \sqrt{c}}{c} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.4

Cancel the common factor of $0.02$ .

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Step 1.4.1

Factor $0.02$ out of $0.02 \sqrt{c}$ .

$\frac{0.02 (\sqrt{c})}{c} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.4.2

Cancel the common factor.

$\frac{0.02 \sqrt{c}}{c} \cdot 0.1 = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.4.3

Rewrite the expression.

$\frac{\sqrt{c}}{c} \cdot \frac{0.1}{50} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.5

Multiply $\frac{\sqrt{c}}{c}$ by $\frac{0.1}{50}$ .

$\frac{\sqrt{c} \cdot 0.1}{c \cdot 50} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.6

Move $0.1$ to the left of $\sqrt{c}$ .

$\frac{0.1 \cdot \sqrt{c}}{c \cdot 50} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.7

Move $50$ to the left of $c$ .

$\frac{0.1 \cdot \sqrt{c}}{50 \cdot c} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.8

Factor $0.1$ out of $0.1 \cdot \sqrt{c}$ .

$\frac{0.1 (\sqrt{c})}{50 \cdot c} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.9

Factor $50$ out of $50 \cdot c$ .

$\frac{0.1 (\sqrt{c})}{50 (c)} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.10

Separate fractions.

$\frac{0.1}{50} \cdot \frac{\sqrt{c}}{c} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.11

Divide $0.1$ by $50$ .

$0.002 \frac{\sqrt{c}}{c} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.12

Combine $0.002$ and $\frac{\sqrt{c}}{c}$ .

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4}{\sqrt{w}}$

Step 1.13

Multiply $\frac{0.4}{\sqrt{w}}$ by $\frac{\sqrt{w}}{\sqrt{w}}$ .

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot (\frac{0.4}{\sqrt{w}} \cdot \frac{\sqrt{w}}{\sqrt{w}})$

Step 1.14

Combine and simplify the denominator.

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Step 1.14.1

Multiply $\frac{0.4}{\sqrt{w}}$ by $\frac{\sqrt{w}}{\sqrt{w}}$ .

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{\sqrt{w} \sqrt{w}}$

Step 1.14.2

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

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{{\sqrt{w}}^{1} \sqrt{w}}$

Step 1.14.3

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

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{{\sqrt{w}}^{1} {\sqrt{w}}^{1}}$

Step 1.14.4

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

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{{\sqrt{w}}^{1 + 1}}$

Step 1.14.5

Add $1$ and $1$ .

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{{\sqrt{w}}^{2}}$

Step 1.14.6

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

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Step 1.14.6.1

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

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{{(w^{\frac{1}{2}})}^{2}}$

Step 1.14.6.2

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

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{w^{\frac{1}{2} \cdot 2}}$

Step 1.14.6.3

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

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{w^{\frac{2}{2}}}$

Step 1.14.6.4

Cancel the common factor of $2$ .

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Step 1.14.6.4.1

Cancel the common factor.

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{w^{\frac{2}{2}}}$

Step 1.14.6.4.2

Rewrite the expression.

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{w^{1}}$

Step 1.14.6.5

Simplify.

$\frac{0.002 \sqrt{c}}{c} = - 0.8 \cdot \frac{0.4 \sqrt{w}}{w}$

Step 1.15

Multiply $- 0.8 \frac{0.4 \sqrt{w}}{w}$ .

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Step 1.15.1

Combine $- 0.8$ and $\frac{0.4 \sqrt{w}}{w}$ .

$\frac{0.002 \sqrt{c}}{c} = \frac{- 0.8 (0.4 \sqrt{w})}{w}$

Step 1.15.2

Multiply $0.4$ by $- 0.8$ .

$\frac{0.002 \sqrt{c}}{c} = \frac{- 0.32 \sqrt{w}}{w}$

Step 1.16

Move the negative in front of the fraction.

$\frac{0.002 \sqrt{c}}{c} = - \frac{0.32 \sqrt{w}}{w}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$0.002 \sqrt{c} w = c (- 0.32 \sqrt{w})$

Step 3

Solve the equation for $c$ .

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Step 3.1

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

${(0.002 \sqrt{c} w)}^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2

Simplify each side of the equation.

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Step 3.2.1

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

${(0.002 c^{\frac{1}{2}} w)}^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify ${(0.002 c^{\frac{1}{2}} w)}^{2}$ .

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Step 3.2.2.1.1

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

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Step 3.2.2.1.1.1

Apply the product rule to $0.002 c^{\frac{1}{2}} w$ .

${(0.002 c^{\frac{1}{2}})}^{2} w^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2.2.1.1.2

Apply the product rule to $0.002 c^{\frac{1}{2}}$ .

${0.002}^{2} {(c^{\frac{1}{2}})}^{2} w^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2.2.1.2

Raise $0.002$ to the power of $2$ .

$0.000004 {(c^{\frac{1}{2}})}^{2} w^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2.2.1.3

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

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Step 3.2.2.1.3.1

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

$0.000004 c^{\frac{1}{2} \cdot 2} w^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.2.1

Cancel the common factor.

$0.000004 c^{\frac{1}{2} \cdot 2} w^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2.2.1.3.2.2

Rewrite the expression.

$0.000004 c^{1} w^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2.2.1.4

Simplify.

$0.000004 c w^{2} = {(c (- 0.32 \sqrt{w}))}^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify ${(c (- 0.32 \sqrt{w}))}^{2}$ .

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Step 3.2.3.1.1

Rewrite using the commutative property of multiplication.

$0.000004 c w^{2} = {(- 0.32 c \sqrt{w})}^{2}$

Step 3.2.3.1.2

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

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Step 3.2.3.1.2.1

Apply the product rule to $- 0.32 c \sqrt{w}$ .

$0.000004 c w^{2} = {(- 0.32 c)}^{2} {\sqrt{w}}^{2}$

Step 3.2.3.1.2.2

Apply the product rule to $- 0.32 c$ .

$0.000004 c w^{2} = {(- 0.32)}^{2} c^{2} {\sqrt{w}}^{2}$

Step 3.2.3.1.3

Raise $- 0.32$ to the power of $2$ .

$0.000004 c w^{2} = 0.1024 c^{2} {\sqrt{w}}^{2}$

Step 3.2.3.1.4

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

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Step 3.2.3.1.4.1

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

$0.000004 c w^{2} = 0.1024 c^{2} {(w^{\frac{1}{2}})}^{2}$

Step 3.2.3.1.4.2

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

$0.000004 c w^{2} = 0.1024 c^{2} w^{\frac{1}{2} \cdot 2}$

Step 3.2.3.1.4.3

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

$0.000004 c w^{2} = 0.1024 c^{2} w^{\frac{2}{2}}$

Step 3.2.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.2.3.1.4.4.1

Cancel the common factor.

$0.000004 c w^{2} = 0.1024 c^{2} w^{\frac{2}{2}}$

Step 3.2.3.1.4.4.2

Rewrite the expression.

$0.000004 c w^{2} = 0.1024 c^{2} w^{1}$

Step 3.2.3.1.4.5

Simplify.

$0.000004 c w^{2} = 0.1024 c^{2} w$

Step 3.3

Solve for $c$ .

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Step 3.3.1

Subtract $0.1024 c^{2} w$ from both sides of the equation.

$0.000004 c w^{2} - 0.1024 c^{2} w = 0$

Step 3.3.2

Factor $0.000004 c w$ out of $0.000004 c w^{2} - 0.1024 c^{2} w$ .

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Step 3.3.2.1

Factor $0.000004 c w$ out of $0.000004 c w^{2}$ .

$0.000004 c w (w) - 0.1024 c^{2} w = 0$

Step 3.3.2.2

Factor $0.000004 c w$ out of $- 0.1024 c^{2} w$ .

$0.000004 c w (w) + 0.000004 c w (- 25600 c) = 0$

Step 3.3.2.3

Factor $0.000004 c w$ out of $0.000004 c w (w) + 0.000004 c w (- 25600 c)$ .

$0.000004 c w (w - 25600 c) = 0$

Step 3.3.3

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

$c = 0$

$w - 25600 c = 0$

Step 3.3.4

Set $c$ equal to $0$ .

$c = 0$

Step 3.3.5

Set $w - 25600 c$ equal to $0$ and solve for $c$ .

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Step 3.3.5.1

Set $w - 25600 c$ equal to $0$ .

$w - 25600 c = 0$

Step 3.3.5.2

Solve $w - 25600 c = 0$ for $c$ .

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Step 3.3.5.2.1

Subtract $w$ from both sides of the equation.

$- 25600 c = - w$

Step 3.3.5.2.2

Divide each term in $- 25600 c = - w$ by $- 25600$ and simplify.

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Step 3.3.5.2.2.1

Divide each term in $- 25600 c = - w$ by $- 25600$ .

$\frac{- 25600 c}{- 25600} = \frac{- w}{- 25600}$

Step 3.3.5.2.2.2

Simplify the left side.

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Step 3.3.5.2.2.2.1

Cancel the common factor of $- 25600$ .

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Step 3.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 25600 c}{- 25600} = \frac{- w}{- 25600}$

Step 3.3.5.2.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{- w}{- 25600}$

Step 3.3.5.2.2.3

Simplify the right side.

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Step 3.3.5.2.2.3.1

Dividing two negative values results in a positive value.

$c = \frac{w}{25600}$

Step 3.3.6

The final solution is all the values that make $0.000004 c w (w - 25600 c) = 0$ true.

$c = 0$

$c = \frac{w}{25600}$

$c = 0$

$c = \frac{w}{25600}$

$c = 0$

$c = \frac{w}{25600}$