Basic Math Examples

r \cdot (R - H) = R \sqrt{H^{2} + r^{2}}

$r \cdot (R - H) = R \sqrt{H^{2} + r^{2}}$

Step 1

Solve for

R \sqrt{H^{2} + r^{2}}

$R \sqrt{H^{2} + r^{2}}$ .

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Step 1.1

Rewrite the equation as

R \sqrt{H^{2} + r^{2}} = r \cdot (R - H)

$R \sqrt{H^{2} + r^{2}} = r \cdot (R - H)$ .

R \sqrt{H^{2} + r^{2}} = r \cdot (R - H)

$R \sqrt{H^{2} + r^{2}} = r \cdot (R - H)$

Step 1.2

Simplify

r \cdot (R - H)

$r \cdot (R - H)$ .

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Step 1.2.1

Apply the distributive property.

R \sqrt{H^{2} + r^{2}} = r R + r (- H)

$R \sqrt{H^{2} + r^{2}} = r R + r (- H)$

Step 1.2.2

Rewrite using the commutative property of multiplication.

R \sqrt{H^{2} + r^{2}} = r R - r H

$R \sqrt{H^{2} + r^{2}} = r R - r H$

R \sqrt{H^{2} + r^{2}} = r R - r H

$R \sqrt{H^{2} + r^{2}} = r R - r H$

R \sqrt{H^{2} + r^{2}} = r R - r H

$R \sqrt{H^{2} + r^{2}} = r R - r H$

Step 2

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

{(R \sqrt{H^{2} + r^{2}})}^{2} = {(r R - r H)}^{2}

${(R \sqrt{H^{2} + r^{2}})}^{2} = {(r R - r H)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{H^{2} + r^{2}}

$\sqrt{H^{2} + r^{2}}$ as

{(H^{2} + r^{2})}^{\frac{1}{2}}

${(H^{2} + r^{2})}^{\frac{1}{2}}$ .

{(R {(H^{2} + r^{2})}^{\frac{1}{2}})}^{2} = {(r R - r H)}^{2}

${(R {(H^{2} + r^{2})}^{\frac{1}{2}})}^{2} = {(r R - r H)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify

{(R {(H^{2} + r^{2})}^{\frac{1}{2}})}^{2}

${(R {(H^{2} + r^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to

R {(H^{2} + r^{2})}^{\frac{1}{2}}

$R {(H^{2} + r^{2})}^{\frac{1}{2}}$ .

R^{2} {({(H^{2} + r^{2})}^{\frac{1}{2}})}^{2} = {(r R - r H)}^{2}

$R^{2} {({(H^{2} + r^{2})}^{\frac{1}{2}})}^{2} = {(r R - r H)}^{2}$

Step 3.2.1.2

Multiply the exponents in

{({(H^{2} + r^{2})}^{\frac{1}{2}})}^{2}

${({(H^{2} + r^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

R^{2} {(H^{2} + r^{2})}^{\frac{1}{2} \cdot 2} = {(r R - r H)}^{2}

$R^{2} {(H^{2} + r^{2})}^{\frac{1}{2} \cdot 2} = {(r R - r H)}^{2}$

Step 3.2.1.2.2

Cancel the common factor of

2

$2$ .

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Step 3.2.1.2.2.1

Cancel the common factor.

$R^{2} {(H^{2} + r^{2})}^{\frac{1}{2} \cdot 2} = {(r R - r H)}^{2}$

Step 3.2.1.2.2.2

Rewrite the expression.

$R^{2} {(H^{2} + r^{2})}^{1} = {(r R - r H)}^{2}$

Step 3.2.1.3

Simplify.

$R^{2} (H^{2} + r^{2}) = {(r R - r H)}^{2}$

Step 3.2.1.4

Apply the distributive property.

$R^{2} H^{2} + R^{2} r^{2} = {(r R - r H)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify

${(r R - r H)}^{2}$ .

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Step 3.3.1.1

Rewrite

${(r R - r H)}^{2}$ as

$(r R - r H) (r R - r H)$ .

$R^{2} H^{2} + R^{2} r^{2} = (r R - r H) (r R - r H)$

Step 3.3.1.2

Expand

$(r R - r H) (r R - r H)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$R^{2} H^{2} + R^{2} r^{2} = r R (r R - r H) - r H (r R - r H)$

Step 3.3.1.2.2

Apply the distributive property.

$R^{2} H^{2} + R^{2} r^{2} = r R (r R) + r R (- r H) - r H (r R - r H)$

Step 3.3.1.2.3

Apply the distributive property.

$R^{2} H^{2} + R^{2} r^{2} = r R (r R) + r R (- r H) - r H (r R) - r H (- r H)$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply

$r$ by

$r$ by adding the exponents.

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Step 3.3.1.3.1.1.1

Move

$r$ .

$R^{2} H^{2} + R^{2} r^{2} = r \cdot r R \cdot R + r R (- r H) - r H (r R) - r H (- r H)$

Step 3.3.1.3.1.1.2

Multiply

$r$ by

$r$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R \cdot R + r R (- r H) - r H (r R) - r H (- r H)$

Step 3.3.1.3.1.2

Multiply

$R$ by

$R$ by adding the exponents.

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Step 3.3.1.3.1.2.1

Move

$R$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} (R \cdot R) + r R (- r H) - r H (r R) - r H (- r H)$

Step 3.3.1.3.1.2.2

Multiply

$R$ by

$R$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} + r R (- r H) - r H (r R) - r H (- r H)$

Step 3.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r R (r H) - r H (r R) - r H (- r H)$

Step 3.3.1.3.1.4

Multiply

$r$ by

$r$ by adding the exponents.

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Step 3.3.1.3.1.4.1

Move

$r$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - (r \cdot r) R H - r H (r R) - r H (- r H)$

Step 3.3.1.3.1.4.2

Multiply

$r$ by

$r$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - r H (r R) - r H (- r H)$

Step 3.3.1.3.1.5

Multiply

$r$ by

$r$ by adding the exponents.

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Step 3.3.1.3.1.5.1

Move

$r$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - (r \cdot r) H R - r H (- r H)$

Step 3.3.1.3.1.5.2

Multiply

$r$ by

$r$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - r^{2} H R - r H (- r H)$

Step 3.3.1.3.1.6

Multiply

$r$ by

$r$ by adding the exponents.

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Step 3.3.1.3.1.6.1

Move

$r$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - r^{2} H R - (r \cdot r) H (- H)$

Step 3.3.1.3.1.6.2

Multiply

$r$ by

$r$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - r^{2} H R - r^{2} H (- H)$

Step 3.3.1.3.1.7

Multiply

$H$ by

$H$ by adding the exponents.

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Step 3.3.1.3.1.7.1

Move

$H$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - r^{2} H R - r^{2} (H \cdot H) \cdot - 1$

Step 3.3.1.3.1.7.2

Multiply

$H$ by

$H$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - r^{2} H R - r^{2} H^{2} \cdot - 1$

Step 3.3.1.3.1.8

Multiply

$- r^{2} H^{2} \cdot - 1$ .

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Step 3.3.1.3.1.8.1

Multiply

$- 1$ by

$- 1$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - r^{2} H R + 1 r^{2} H^{2}$

Step 3.3.1.3.1.8.2

Multiply

$r^{2}$ by

$1$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} R H - r^{2} H R + r^{2} H^{2}$

Step 3.3.1.3.2

Subtract

$r^{2} H R$ from

$- r^{2} R H$ .

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Step 3.3.1.3.2.1

Move

$R$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - r^{2} H R - r^{2} H R + r^{2} H^{2}$

Step 3.3.1.3.2.2

Subtract

$r^{2} H R$ from

$- r^{2} H R$ .

$R^{2} H^{2} + R^{2} r^{2} = r^{2} R^{2} - 2 r^{2} H R + r^{2} H^{2}$

Step 4

Solve for

$H$ .

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Step 4.1

Since

$H$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$r^{2} R^{2} - 2 r^{2} H R + r^{2} H^{2} = R^{2} H^{2} + R^{2} r^{2}$

Step 4.2

Subtract

$R^{2} H^{2}$ from both sides of the equation.

$r^{2} R^{2} - 2 r^{2} H R + r^{2} H^{2} - R^{2} H^{2} = R^{2} r^{2}$

Step 4.3

Move all terms not containing

$H$ to the right side of the equation.

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Step 4.3.1

Subtract

$r^{2} R^{2}$ from both sides of the equation.

$- 2 r^{2} H R + r^{2} H^{2} - R^{2} H^{2} = R^{2} r^{2} - r^{2} R^{2}$

Step 4.3.2

Combine the opposite terms in

$R^{2} r^{2} - r^{2} R^{2}$ .

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Step 4.3.2.1

Reorder the factors in the terms

$R^{2} r^{2}$ and

$- r^{2} R^{2}$ .

$- 2 r^{2} H R + r^{2} H^{2} - R^{2} H^{2} = R^{2} r^{2} - R^{2} r^{2}$

Step 4.3.2.2

Subtract

$R^{2} r^{2}$ from

$R^{2} r^{2}$ .

$- 2 r^{2} H R + r^{2} H^{2} - R^{2} H^{2} = 0$

Step 4.4

Factor

$H$ out of

$- 2 r^{2} H R + r^{2} H^{2} - R^{2} H^{2}$ .

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Step 4.4.1

Factor

$H$ out of

$- 2 r^{2} H R$ .

$H (- 2 r^{2} R) + r^{2} H^{2} - R^{2} H^{2} = 0$

Step 4.4.2

Factor

$H$ out of

$r^{2} H^{2}$ .

$H (- 2 r^{2} R) + H (r^{2} H) - R^{2} H^{2} = 0$

Step 4.4.3

Factor

$H$ out of

$- R^{2} H^{2}$ .

$H (- 2 r^{2} R) + H (r^{2} H) + H (- R^{2} H) = 0$

Step 4.4.4

Factor

$H$ out of

$H (- 2 r^{2} R) + H (r^{2} H)$ .

$H (- 2 r^{2} R + r^{2} H) + H (- R^{2} H) = 0$

Step 4.4.5

Factor

$H$ out of

$H (- 2 r^{2} R + r^{2} H) + H (- R^{2} H)$ .

$H (- 2 r^{2} R + r^{2} H - R^{2} H) = 0$

Step 4.5

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$H = 0$

$- 2 r^{2} R + r^{2} H - R^{2} H = 0$

Step 4.6

Set

$H$ equal to

$0$ .

$H = 0$

Step 4.7

Set

$- 2 r^{2} R + r^{2} H - R^{2} H$ equal to

$0$ and solve for

$H$ .

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Step 4.7.1

Set

$- 2 r^{2} R + r^{2} H - R^{2} H$ equal to

$0$ .

$- 2 r^{2} R + r^{2} H - R^{2} H = 0$

Step 4.7.2

Solve

$- 2 r^{2} R + r^{2} H - R^{2} H = 0$ for

$H$ .

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Step 4.7.2.1

Add

$2 r^{2} R$ to both sides of the equation.

$r^{2} H - R^{2} H = 2 r^{2} R$

Step 4.7.2.2

Factor

$H$ out of

$r^{2} H - R^{2} H$ .

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Step 4.7.2.2.1

Factor

$H$ out of

$r^{2} H$ .

$H r^{2} - R^{2} H = 2 r^{2} R$

Step 4.7.2.2.2

Factor

$H$ out of

$- R^{2} H$ .

$H r^{2} + H (- R^{2}) = 2 r^{2} R$

Step 4.7.2.2.3

Factor

$H$ out of

$H r^{2} + H (- R^{2})$ .

$H (r^{2} - R^{2}) = 2 r^{2} R$

Step 4.7.2.3

Factor.

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Step 4.7.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = r$ and

$b = R$ .

$H ((r + R) (r - R)) = 2 r^{2} R$

Step 4.7.2.3.2

Remove unnecessary parentheses.

$H (r + R) (r - R) = 2 r^{2} R$

Step 4.7.2.4

Divide each term in

$H (r + R) (r - R) = 2 r^{2} R$ by

$(r + R) (r - R)$ and simplify.

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Step 4.7.2.4.1

Divide each term in

$H (r + R) (r - R) = 2 r^{2} R$ by

$(r + R) (r - R)$ .

$\frac{H (r + R) (r - R)}{(r + R) (r - R)} = \frac{2 r^{2} R}{(r + R) (r - R)}$

Step 4.7.2.4.2

Simplify the left side.

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Step 4.7.2.4.2.1

Cancel the common factor of

$r + R$ .

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Step 4.7.2.4.2.1.1

Cancel the common factor.

$\frac{H (r + R) (r - R)}{(r + R) (r - R)} = \frac{2 r^{2} R}{(r + R) (r - R)}$

Step 4.7.2.4.2.1.2

Rewrite the expression.

$\frac{H (r - R)}{r - R} = \frac{2 r^{2} R}{(r + R) (r - R)}$

Step 4.7.2.4.2.2

Cancel the common factor of

$r - R$ .

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Step 4.7.2.4.2.2.1

Cancel the common factor.

$\frac{H (r - R)}{r - R} = \frac{2 r^{2} R}{(r + R) (r - R)}$

Step 4.7.2.4.2.2.2

Divide

$H$ by

$1$ .

$H = \frac{2 r^{2} R}{(r + R) (r - R)}$

Step 4.8

The final solution is all the values that make

$H (- 2 r^{2} R + r^{2} H - R^{2} H) = 0$ true.

$H = 0$

$H = \frac{2 r^{2} R}{(r + R) (r - R)}$

$H = 0$

$H = \frac{2 r^{2} R}{(r + R) (r - R)}$