Calculus Examples

$Q = l^{\frac{1}{2}} k^{\frac{3}{2}} + k^{2}$

Step 1

Rewrite the equation as $l^{\frac{1}{2}} k^{\frac{3}{2}} + k^{2} = Q$ .

$l^{\frac{1}{2}} k^{\frac{3}{2}} + k^{2} = Q$

Step 2

Move all terms not containing $l$ to the right side of the equation.

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Step 2.1

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

$l^{\frac{1}{2}} k^{\frac{3}{2}} - Q = - k^{2}$

Step 2.2

Add $Q$ to both sides of the equation.

$l^{\frac{1}{2}} k^{\frac{3}{2}} = - k^{2} + Q$

Step 3

Divide each term in $l^{\frac{1}{2}} k^{\frac{3}{2}} = - k^{2} + Q$ by $k^{\frac{3}{2}}$ and simplify.

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Step 3.1

Divide each term in $l^{\frac{1}{2}} k^{\frac{3}{2}} = - k^{2} + Q$ by $k^{\frac{3}{2}}$ .

$\frac{l^{\frac{1}{2}} k^{\frac{3}{2}}}{k^{\frac{3}{2}}} = \frac{- k^{2}}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor.

$\frac{l^{\frac{1}{2}} k^{\frac{3}{2}}}{k^{\frac{3}{2}}} = \frac{- k^{2}}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.2.2

Divide $l^{\frac{1}{2}}$ by $1$ .

$l^{\frac{1}{2}} = \frac{- k^{2}}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move $k^{\frac{3}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$l^{\frac{1}{2}} = - k^{2} k^{- \frac{3}{2}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.3.1.2

Multiply $k^{2}$ by $k^{- \frac{3}{2}}$ by adding the exponents.

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Step 3.3.1.2.1

Move $k^{- \frac{3}{2}}$ .

$l^{\frac{1}{2}} = - (k^{- \frac{3}{2}} k^{2}) + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.3.1.2.2

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

$l^{\frac{1}{2}} = - k^{- \frac{3}{2} + 2} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.3.1.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$l^{\frac{1}{2}} = - k^{- \frac{3}{2} + 2 \cdot \frac{2}{2}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.3.1.2.4

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

$l^{\frac{1}{2}} = - k^{- \frac{3}{2} + \frac{2 \cdot 2}{2}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.3.1.2.5

Combine the numerators over the common denominator.

$l^{\frac{1}{2}} = - k^{\frac{- 3 + 2 \cdot 2}{2}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.3.1.2.6

Simplify the numerator.

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Step 3.3.1.2.6.1

Multiply $2$ by $2$ .

$l^{\frac{1}{2}} = - k^{\frac{- 3 + 4}{2}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 3.3.1.2.6.2

Add $- 3$ and $4$ .

$l^{\frac{1}{2}} = - k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}}$

Step 4

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(l^{\frac{1}{2}})}^{2} = {(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})}^{2}$

Step 5

Simplify the exponent.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify ${(l^{\frac{1}{2}})}^{2}$ .

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Step 5.1.1.1

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

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Step 5.1.1.1.1

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

$l^{\frac{1}{2} \cdot 2} = {(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})}^{2}$

Step 5.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.1.1.1.2.1

Cancel the common factor.

$l^{\frac{1}{2} \cdot 2} = {(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})}^{2}$

Step 5.1.1.1.2.2

Rewrite the expression.

$l^{1} = {(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})}^{2}$

Step 5.1.1.2

Simplify.

$l = {(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify ${(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})}^{2}$ .

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Step 5.2.1.1

Rewrite ${(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})}^{2}$ as $(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}}) (- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})$ .

$l = (- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}}) (- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})$

Step 5.2.1.2

Expand $(- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}}) (- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})$ using the FOIL Method.

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Step 5.2.1.2.1

Apply the distributive property.

$l = - k^{\frac{1}{2}} (- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}}) + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})$

Step 5.2.1.2.2

Apply the distributive property.

$l = - k^{\frac{1}{2}} (- k^{\frac{1}{2}}) - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}})$

Step 5.2.1.2.3

Apply the distributive property.

$l = - k^{\frac{1}{2}} (- k^{\frac{1}{2}}) - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3

Simplify and combine like terms.

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Step 5.2.1.3.1

Simplify each term.

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Step 5.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$l = - 1 \cdot - 1 k^{\frac{1}{2}} k^{\frac{1}{2}} - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.2

Multiply $k^{\frac{1}{2}}$ by $k^{\frac{1}{2}}$ by adding the exponents.

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Step 5.2.1.3.1.2.1

Move $k^{\frac{1}{2}}$ .

$l = - 1 \cdot - 1 (k^{\frac{1}{2}} k^{\frac{1}{2}}) - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.2.2

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

$l = - 1 \cdot - 1 k^{\frac{1}{2} + \frac{1}{2}} - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.2.3

Combine the numerators over the common denominator.

$l = - 1 \cdot - 1 k^{\frac{1 + 1}{2}} - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.2.4

Add $1$ and $1$ .

$l = - 1 \cdot - 1 k^{\frac{2}{2}} - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.2.5

Divide $2$ by $2$ .

$l = - 1 \cdot - 1 k^{1} - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.3

Simplify $- 1 \cdot - 1 k^{1}$ .

$l = - 1 \cdot - 1 k - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.4

Multiply $- 1$ by $- 1$ .

$l = 1 k - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.5

Multiply $k$ by $1$ .

$l = k - k^{\frac{1}{2}} \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.6

Cancel the common factor of $k^{\frac{1}{2}}$ .

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Step 5.2.1.3.1.6.1

Factor $k^{\frac{1}{2}}$ out of $- k^{\frac{1}{2}}$ .

$l = k + k^{\frac{1}{2}} \cdot - 1 \frac{Q}{k^{\frac{3}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.6.2

Factor $k^{\frac{1}{2}}$ out of $k^{\frac{3}{2}}$ .

$l = k + k^{\frac{1}{2}} \cdot - 1 \frac{Q}{k^{\frac{1}{2}} k^{\frac{2}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.6.3

Cancel the common factor.

$l = k + k^{\frac{1}{2}} \cdot - 1 \frac{Q}{k^{\frac{1}{2}} k^{\frac{2}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.6.4

Rewrite the expression.

$l = k - 1 \frac{Q}{k^{\frac{2}{2}}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.7

Simplify the denominator.

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Step 5.2.1.3.1.7.1

Divide $2$ by $2$ .

$l = k - 1 \frac{Q}{k^{1}} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.7.2

Simplify.

$l = k - 1 \frac{Q}{k} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.8

Rewrite $- 1 \frac{Q}{k}$ as $- \frac{Q}{k}$ .

$l = k - \frac{Q}{k} + \frac{Q}{k^{\frac{3}{2}}} (- k^{\frac{1}{2}}) + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.9

Rewrite using the commutative property of multiplication.

$l = k - \frac{Q}{k} - \frac{Q}{k^{\frac{3}{2}}} k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.10

Cancel the common factor of $k^{\frac{1}{2}}$ .

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Step 5.2.1.3.1.10.1

Move the leading negative in $- \frac{Q}{k^{\frac{3}{2}}}$ into the numerator.

$l = k - \frac{Q}{k} + \frac{- Q}{k^{\frac{3}{2}}} k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.10.2

Factor $k^{\frac{1}{2}}$ out of $k^{\frac{3}{2}}$ .

$l = k - \frac{Q}{k} + \frac{- Q}{k^{\frac{1}{2}} k^{\frac{2}{2}}} k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.10.3

Cancel the common factor.

$l = k - \frac{Q}{k} + \frac{- Q}{k^{\frac{1}{2}} k^{\frac{2}{2}}} k^{\frac{1}{2}} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.10.4

Rewrite the expression.

$l = k - \frac{Q}{k} + \frac{- Q}{k^{\frac{2}{2}}} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.11

Cancel the common factor of $2$ .

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Step 5.2.1.3.1.11.1

Cancel the common factor.

$l = k - \frac{Q}{k} + \frac{- Q}{k^{\frac{2}{2}}} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.11.2

Rewrite the expression.

$l = k - \frac{Q}{k} + \frac{- Q}{k^{1}} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.12

Simplify.

$l = k - \frac{Q}{k} + \frac{- Q}{k} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.13

Move the negative in front of the fraction.

$l = k - \frac{Q}{k} - \frac{Q}{k} + \frac{Q}{k^{\frac{3}{2}}} \cdot \frac{Q}{k^{\frac{3}{2}}}$

Step 5.2.1.3.1.14

Combine.

$l = k - \frac{Q}{k} - \frac{Q}{k} + \frac{Q \cdot Q}{k^{\frac{3}{2}} k^{\frac{3}{2}}}$

Step 5.2.1.3.1.15

Multiply $Q$ by $Q$ .

$l = k - \frac{Q}{k} - \frac{Q}{k} + \frac{Q^{2}}{k^{\frac{3}{2}} k^{\frac{3}{2}}}$

Step 5.2.1.3.1.16

Multiply $k^{\frac{3}{2}}$ by $k^{\frac{3}{2}}$ by adding the exponents.

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Step 5.2.1.3.1.16.1

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

$l = k - \frac{Q}{k} - \frac{Q}{k} + \frac{Q^{2}}{k^{\frac{3}{2} + \frac{3}{2}}}$

Step 5.2.1.3.1.16.2

Combine the numerators over the common denominator.

$l = k - \frac{Q}{k} - \frac{Q}{k} + \frac{Q^{2}}{k^{\frac{3 + 3}{2}}}$

Step 5.2.1.3.1.16.3

Add $3$ and $3$ .

$l = k - \frac{Q}{k} - \frac{Q}{k} + \frac{Q^{2}}{k^{\frac{6}{2}}}$

Step 5.2.1.3.1.16.4

Divide $6$ by $2$ .

$l = k - \frac{Q}{k} - \frac{Q}{k} + \frac{Q^{2}}{k^{3}}$

Step 5.2.1.3.2

Subtract $\frac{Q}{k}$ from $- \frac{Q}{k}$ .

$l = k - 2 \frac{Q}{k} + \frac{Q^{2}}{k^{3}}$

Step 5.2.1.4

Simplify each term.

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Step 5.2.1.4.1

Combine $- 2$ and $\frac{Q}{k}$ .

$l = k + \frac{- 2 Q}{k} + \frac{Q^{2}}{k^{3}}$

Step 5.2.1.4.2

Move the negative in front of the fraction.

$l = k - \frac{2 Q}{k} + \frac{Q^{2}}{k^{3}}$

Step 6

Simplify $k - \frac{2 Q}{k} + \frac{Q^{2}}{k^{3}}$ .

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Step 6.1

Move $- \frac{2 Q}{k}$ .

$l = k + \frac{Q^{2}}{k^{3}} - \frac{2 Q}{k}$

Step 6.2

Reorder $k$ and $\frac{Q^{2}}{k^{3}}$ .

$l = \frac{Q^{2}}{k^{3}} + k - \frac{2 Q}{k}$