Algebra Examples

$v = 1.34 \sqrt{l}$

Step 1

Interchange the variables.

$l = 1.34 \sqrt{y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $1.34 \sqrt{y} = l$ .

$1.34 \sqrt{y} = l$

Step 2.2

Divide each term in $1.34 \sqrt{y} = l$ by $1.34$ and simplify.

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Step 2.2.1

Divide each term in $1.34 \sqrt{y} = l$ by $1.34$ .

$\frac{1.34 \sqrt{y}}{1.34} = \frac{l}{1.34}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $1.34$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{1.34 \sqrt{y}}{1.34} = \frac{l}{1.34}$

Step 2.2.2.1.2

Divide $\sqrt{y}$ by $1$ .

$\sqrt{y} = \frac{l}{1.34}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply by $1$ .

$\sqrt{y} = \frac{1 l}{1.34}$

Step 2.2.3.2

Factor $1.34$ out of $1.34$ .

$\sqrt{y} = \frac{1 l}{1.34 (1)}$

Step 2.2.3.3

Separate fractions.

$\sqrt{y} = \frac{1}{1.34} \cdot \frac{l}{1}$

Step 2.2.3.4

Divide $1$ by $1.34$ .

$\sqrt{y} = 0.74626865 \frac{l}{1}$

Step 2.2.3.5

Divide $l$ by $1$ .

$\sqrt{y} = 0.74626865 l$

Step 2.3

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

${\sqrt{y}}^{2} = {(0.74626865 l)}^{2}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

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

${(y^{\frac{1}{2}})}^{2} = {(0.74626865 l)}^{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

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

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Step 2.4.2.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(0.74626865 l)}^{2}$

Step 2.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(0.74626865 l)}^{2}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(0.74626865 l)}^{2}$

Step 2.4.2.1.2

Simplify.

$y = {(0.74626865 l)}^{2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(0.74626865 l)}^{2}$ .

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Step 2.4.3.1.1

Apply the product rule to $0.74626865 l$ .

$y = {0.74626865}^{2} l^{2}$

Step 2.4.3.1.2

Raise $0.74626865$ to the power of $2$ .

$y = 0.5569169 l^{2}$

Step 3

Replace $y$ with $f^{- 1} (l)$ to show the final answer.

$f^{- 1} (l) = 0.5569169 l^{2}$

Step 4

Verify if $f^{- 1} (l) = 0.5569169 l^{2}$ is the inverse of $f (l) = 1.34 \sqrt{l}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (l)) = l$ and $f (f^{- 1} (l)) = l$ .

Step 4.2

Evaluate $f^{- 1} (f (l))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (l))$

Step 4.2.2

Evaluate $f^{- 1} (1.34 \sqrt{l})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 {(1.34 \sqrt{l})}^{2}$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Apply the product rule to $1.34 \sqrt{l}$ .

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 ({1.34}^{2} {\sqrt{l}}^{2})$

Step 4.2.3.2

Raise $1.34$ to the power of $2$ .

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 (1.7956 {\sqrt{l}}^{2})$

Step 4.2.4

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

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Step 4.2.4.1

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

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 (1.7956 {(l^{\frac{1}{2}})}^{2})$

Step 4.2.4.2

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

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 (1.7956 l^{\frac{1}{2} \cdot 2})$

Step 4.2.4.3

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

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 (1.7956 l^{\frac{2}{2}})$

Step 4.2.4.4

Cancel the common factor of $2$ .

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Step 4.2.4.4.1

Cancel the common factor.

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 (1.7956 l^{\frac{2}{2}})$

Step 4.2.4.4.2

Rewrite the expression.

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 (1.7956 l)$

Step 4.2.4.5

Simplify.

$f^{- 1} (1.34 \sqrt{l}) = 0.5569169 (1.7956 l)$

Step 4.2.5

Multiply $1.7956$ by $0.5569169$ .

$f^{- 1} (1.34 \sqrt{l}) = 1 l$

Step 4.3

Evaluate $f (f^{- 1} (l))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (l))$

Step 4.3.2

Evaluate $f (0.5569169 l^{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (0.5569169 l^{2}) = 1.34 \sqrt{0.5569169 l^{2}}$

Step 4.3.3

Remove parentheses.

$f (0.5569169 l^{2}) = 1.34 \sqrt{0.5569169 l^{2}}$

Step 4.3.4

Rewrite $0.5569169 l^{2}$ as ${(0.74626865 l)}^{2}$ .

$f (0.5569169 l^{2}) = 1.34 \sqrt{{(0.74626865 l)}^{2}}$

Step 4.3.5

Pull terms out from under the radical, assuming positive real numbers.

$f (0.5569169 l^{2}) = 1.34 (0.74626865 l)$

Step 4.3.6

Multiply $0.74626865$ by $1.34$ .

$f (0.5569169 l^{2}) = 1 l$

Step 4.4

Since $f^{- 1} (f (l)) = l$ and $f (f^{- 1} (l)) = l$ , then $f^{- 1} (l) = 0.5569169 l^{2}$ is the inverse of $f (l) = 1.34 \sqrt{l}$ .

$f^{- 1} (l) = 0.5569169 l^{2}$