Algebra Examples

$d = 0.2 t^{\frac{3}{2}}$

Step 1

Interchange the variables.

$t = 0.2 y^{\frac{3}{2}}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $0.2 y^{\frac{3}{2}} = t$ .

$0.2 y^{\frac{3}{2}} = t$

Step 2.2

Divide each term in $0.2 y^{\frac{3}{2}} = t$ by $0.2$ and simplify.

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Step 2.2.1

Divide each term in $0.2 y^{\frac{3}{2}} = t$ by $0.2$ .

$\frac{0.2 y^{\frac{3}{2}}}{0.2} = \frac{t}{0.2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor.

$\frac{0.2 y^{\frac{3}{2}}}{0.2} = \frac{t}{0.2}$

Step 2.2.2.2

Divide $y^{\frac{3}{2}}$ by $1$ .

$y^{\frac{3}{2}} = \frac{t}{0.2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply by $1$ .

$y^{\frac{3}{2}} = \frac{1 t}{0.2}$

Step 2.2.3.2

Factor $0.2$ out of $0.2$ .

$y^{\frac{3}{2}} = \frac{1 t}{0.2 (1)}$

Step 2.2.3.3

Separate fractions.

$y^{\frac{3}{2}} = \frac{1}{0.2} \cdot \frac{t}{1}$

Step 2.2.3.4

Divide $1$ by $0.2$ .

$y^{\frac{3}{2}} = 5 \frac{t}{1}$

Step 2.2.3.5

Divide $t$ by $1$ .

$y^{\frac{3}{2}} = 5 t$

Step 2.3

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{3}{2}})}^{\frac{2}{3}} = {(5 t)}^{\frac{2}{3}}$

Step 2.4

Simplify the exponent.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.4.1.1.1

Multiply the exponents in ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.4.1.1.1.1

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

$y^{\frac{3}{2} \cdot \frac{2}{3}} = {(5 t)}^{\frac{2}{3}}$

Step 2.4.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.4.1.1.1.2.1

Cancel the common factor.

$y^{\frac{3}{2} \cdot \frac{2}{3}} = {(5 t)}^{\frac{2}{3}}$

Step 2.4.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{2} \cdot 2} = {(5 t)}^{\frac{2}{3}}$

Step 2.4.1.1.1.3

Cancel the common factor of $2$ .

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Step 2.4.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(5 t)}^{\frac{2}{3}}$

Step 2.4.1.1.1.3.2

Rewrite the expression.

$y^{1} = {(5 t)}^{\frac{2}{3}}$

Step 2.4.1.1.2

Simplify.

$y = {(5 t)}^{\frac{2}{3}}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Apply the product rule to $5 t$ .

$y = 5^{\frac{2}{3}} t^{\frac{2}{3}}$

Step 3

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

$f^{- 1} (t) = 5^{\frac{2}{3}} t^{\frac{2}{3}}$

Step 4

Verify if $f^{- 1} (t) = 5^{\frac{2}{3}} t^{\frac{2}{3}}$ is the inverse of $f (t) = 0.2 t^{\frac{3}{2}}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (0.2 t^{\frac{3}{2}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} {(0.2 t^{\frac{3}{2}})}^{\frac{2}{3}}$

Step 4.2.3

Apply the product rule to $0.2 t^{\frac{3}{2}}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} ({0.2}^{\frac{2}{3}} {(t^{\frac{3}{2}})}^{\frac{2}{3}})$

Step 4.2.4

Raise $0.2$ to the power of $\frac{2}{3}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 {(t^{\frac{3}{2}})}^{\frac{2}{3}})$

Step 4.2.5

Multiply the exponents in ${(t^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 4.2.5.1

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

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t^{\frac{3}{2} \cdot \frac{2}{3}})$

Step 4.2.5.2

Cancel the common factor of $3$ .

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Step 4.2.5.2.1

Cancel the common factor.

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t^{\frac{3}{2} \cdot \frac{2}{3}})$

Step 4.2.5.2.2

Rewrite the expression.

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t^{\frac{1}{2} \cdot 2})$

Step 4.2.5.3

Cancel the common factor of $2$ .

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Step 4.2.5.3.1

Cancel the common factor.

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t^{\frac{1}{2} \cdot 2})$

Step 4.2.5.3.2

Rewrite the expression.

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 5^{\frac{2}{3}} (0.34199518 t)$

Step 4.2.6

Multiply $0.34199518$ by $5^{\frac{2}{3}}$ .

$f^{- 1} (0.2 t^{\frac{3}{2}}) = 1 t$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (5^{\frac{2}{3}} t^{\frac{2}{3}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 {(5^{\frac{2}{3}} t^{\frac{2}{3}})}^{\frac{3}{2}}$

Step 4.3.3

Apply the product rule to $5^{\frac{2}{3}} t^{\frac{2}{3}}$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 ({(5^{\frac{2}{3}})}^{\frac{3}{2}} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Step 4.3.4

Multiply the exponents in ${(5^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 4.3.4.1

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

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5^{\frac{2}{3} \cdot \frac{3}{2}} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Step 4.3.4.2

Cancel the common factor of $2$ .

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Step 4.3.4.2.1

Cancel the common factor.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5^{\frac{2}{3} \cdot \frac{3}{2}} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Step 4.3.4.2.2

Rewrite the expression.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5^{\frac{1}{3} \cdot 3} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Step 4.3.4.3

Cancel the common factor of $3$ .

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Step 4.3.4.3.1

Cancel the common factor.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5^{\frac{1}{3} \cdot 3} {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Step 4.3.4.3.2

Rewrite the expression.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 {(t^{\frac{2}{3}})}^{\frac{3}{2}})$

Step 4.3.5

Multiply the exponents in ${(t^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 4.3.5.1

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

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t^{\frac{2}{3} \cdot \frac{3}{2}})$

Step 4.3.5.2

Cancel the common factor of $2$ .

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Step 4.3.5.2.1

Cancel the common factor.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t^{\frac{2}{3} \cdot \frac{3}{2}})$

Step 4.3.5.2.2

Rewrite the expression.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t^{\frac{1}{3} \cdot 3})$

Step 4.3.5.3

Cancel the common factor of $3$ .

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Step 4.3.5.3.1

Cancel the common factor.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t^{\frac{1}{3} \cdot 3})$

Step 4.3.5.3.2

Rewrite the expression.

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 0.2 (5 t)$

Step 4.3.6

Multiply $0.2 (5 t)$ .

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Step 4.3.6.1

Multiply $5$ by $0.2$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = 1 t$

Step 4.3.6.2

Multiply $t$ by $1$ .

$f (5^{\frac{2}{3}} t^{\frac{2}{3}}) = t$

Step 4.4

Since $f^{- 1} (f (t)) = t$ and $f (f^{- 1} (t)) = t$ , then $f^{- 1} (t) = 5^{\frac{2}{3}} t^{\frac{2}{3}}$ is the inverse of $f (t) = 0.2 t^{\frac{3}{2}}$ .

$f^{- 1} (t) = 5^{\frac{2}{3}} t^{\frac{2}{3}}$