Algebra Examples

$y = x^{\frac{1}{4}} - 6$

Step 1

Interchange the variables.

$x = y^{\frac{1}{4}} - 6$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $y^{\frac{1}{4}} - 6 = x$ .

$y^{\frac{1}{4}} - 6 = x$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Add $6$ to both sides of the equation.

$y^{\frac{1}{4}} - x = 6$

Step 2.2.2

Add $x$ to both sides of the equation.

$y^{\frac{1}{4}} = 6 + x$

Step 2.3

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{4}})}^{4} = {(6 + x)}^{4}$

Step 2.4

Simplify the exponent.

Tap for more steps...

Step 2.4.1

Simplify the left side.

Tap for more steps...

Step 2.4.1.1

Simplify ${(y^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 2.4.1.1.1

Multiply the exponents in ${(y^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 2.4.1.1.1.1

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

$y^{\frac{1}{4} \cdot 4} = {(6 + x)}^{4}$

Step 2.4.1.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.4.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{4} \cdot 4} = {(6 + x)}^{4}$

Step 2.4.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(6 + x)}^{4}$

Step 2.4.1.1.2

Simplify.

$y = {(6 + x)}^{4}$

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Simplify ${(6 + x)}^{4}$ .

Tap for more steps...

Step 2.4.2.1.1

Use the Binomial Theorem.

$y = 6^{4} + 4 \cdot 6^{3} x + 6 \cdot 6^{2} x^{2} + 4 \cdot 6 x^{3} + x^{4}$

Step 2.4.2.1.2

Simplify each term.

Tap for more steps...

Step 2.4.2.1.2.1

Raise $6$ to the power of $4$ .

$y = 1296 + 4 \cdot 6^{3} x + 6 \cdot 6^{2} x^{2} + 4 \cdot 6 x^{3} + x^{4}$

Step 2.4.2.1.2.2

Raise $6$ to the power of $3$ .

$y = 1296 + 4 \cdot 216 x + 6 \cdot 6^{2} x^{2} + 4 \cdot 6 x^{3} + x^{4}$

Step 2.4.2.1.2.3

Multiply $4$ by $216$ .

$y = 1296 + 864 x + 6 \cdot 6^{2} x^{2} + 4 \cdot 6 x^{3} + x^{4}$

Step 2.4.2.1.2.4

Multiply $6$ by $6^{2}$ by adding the exponents.

Tap for more steps...

Step 2.4.2.1.2.4.1

Multiply $6$ by $6^{2}$ .

Tap for more steps...

Step 2.4.2.1.2.4.1.1

Raise $6$ to the power of $1$ .

$y = 1296 + 864 x + 6^{1} \cdot 6^{2} x^{2} + 4 \cdot 6 x^{3} + x^{4}$

Step 2.4.2.1.2.4.1.2

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

$y = 1296 + 864 x + 6^{1 + 2} x^{2} + 4 \cdot 6 x^{3} + x^{4}$

Step 2.4.2.1.2.4.2

Add $1$ and $2$ .

$y = 1296 + 864 x + 6^{3} x^{2} + 4 \cdot 6 x^{3} + x^{4}$

Step 2.4.2.1.2.5

Raise $6$ to the power of $3$ .

$y = 1296 + 864 x + 216 x^{2} + 4 \cdot 6 x^{3} + x^{4}$

Step 2.4.2.1.2.6

Multiply $4$ by $6$ .

$y = 1296 + 864 x + 216 x^{2} + 24 x^{3} + x^{4}$

Step 2.5

Simplify $1296 + 864 x + 216 x^{2} + 24 x^{3} + x^{4}$ .

Tap for more steps...

Step 2.5.1

Move $1296$ .

$y = 864 x + 216 x^{2} + 24 x^{3} + x^{4} + 1296$

Step 2.5.2

Move $864 x$ .

$y = 216 x^{2} + 24 x^{3} + x^{4} + 864 x + 1296$

Step 2.5.3

Move $216 x^{2}$ .

$y = 24 x^{3} + x^{4} + 216 x^{2} + 864 x + 1296$

Step 2.5.4

Reorder $24 x^{3}$ and $x^{4}$ .

$y = x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296$

Step 3

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

$f^{- 1} (x) = x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296$

Step 4

Verify if $f^{- 1} (x) = x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296$ is the inverse of $f (x) = x^{\frac{1}{4}} - 6$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (x^{\frac{1}{4}} - 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = {(x^{\frac{1}{4}} - 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3

Simplify each term.

Tap for more steps...

Step 4.2.3.1

Use the Binomial Theorem.

$f^{- 1} (x^{\frac{1}{4}} - 6) = {(x^{\frac{1}{4}})}^{4} + 4 {(x^{\frac{1}{4}})}^{3} \cdot - 6 + 6 {(x^{\frac{1}{4}})}^{2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2

Simplify each term.

Tap for more steps...

Step 4.2.3.2.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 4.2.3.2.1.1

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

$f^{- 1} (x^{\frac{1}{4}} - 6) = x^{\frac{1}{4} \cdot 4} + 4 {(x^{\frac{1}{4}})}^{3} \cdot - 6 + 6 {(x^{\frac{1}{4}})}^{2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.3.2.1.2.1

Cancel the common factor.

Step 4.2.3.2.1.2.2

Rewrite the expression.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 4 {(x^{\frac{1}{4}})}^{3} \cdot - 6 + 6 {(x^{\frac{1}{4}})}^{2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.2

Simplify.

Step 4.2.3.2.3

Multiply the exponents in ${(x^{\frac{1}{4}})}^{3}$ .

Tap for more steps...

Step 4.2.3.2.3.1

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

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 4 x^{\frac{1}{4} \cdot 3} \cdot - 6 + 6 {(x^{\frac{1}{4}})}^{2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.3.2

Combine $\frac{1}{4}$ and $3$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 4 x^{\frac{3}{4}} \cdot - 6 + 6 {(x^{\frac{1}{4}})}^{2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.4

Multiply $- 6$ by $4$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 6 {(x^{\frac{1}{4}})}^{2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.5

Multiply the exponents in ${(x^{\frac{1}{4}})}^{2}$ .

Tap for more steps...

Step 4.2.3.2.5.1

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

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 6 x^{\frac{1}{4} \cdot 2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.3.2.5.2.1

Factor $2$ out of $4$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 6 x^{\frac{1}{2 (2)} \cdot 2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.5.2.2

Cancel the common factor.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 6 x^{\frac{1}{2 \cdot 2} \cdot 2} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.5.2.3

Rewrite the expression.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 6 x^{\frac{1}{2}} {(- 6)}^{2} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.6

Raise $- 6$ to the power of $2$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 6 x^{\frac{1}{2}} \cdot 36 + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.7

Multiply $36$ by $6$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} + 4 x^{\frac{1}{4}} {(- 6)}^{3} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.8

Raise $- 6$ to the power of $3$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} + 4 x^{\frac{1}{4}} \cdot - 216 + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.9

Multiply $- 216$ by $4$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + {(- 6)}^{4} + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.2.10

Raise $- 6$ to the power of $4$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 {(x^{\frac{1}{4}} - 6)}^{3} + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.3

Use the Binomial Theorem.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 ({(x^{\frac{1}{4}})}^{3} + 3 {(x^{\frac{1}{4}})}^{2} \cdot - 6 + 3 x^{\frac{1}{4}} {(- 6)}^{2} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4

Simplify each term.

Tap for more steps...

Step 4.2.3.4.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{3}$ .

Tap for more steps...

Step 4.2.3.4.1.1

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

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{1}{4} \cdot 3} + 3 {(x^{\frac{1}{4}})}^{2} \cdot - 6 + 3 x^{\frac{1}{4}} {(- 6)}^{2} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.1.2

Combine $\frac{1}{4}$ and $3$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} + 3 {(x^{\frac{1}{4}})}^{2} \cdot - 6 + 3 x^{\frac{1}{4}} {(- 6)}^{2} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.2

Multiply the exponents in ${(x^{\frac{1}{4}})}^{2}$ .

Tap for more steps...

Step 4.2.3.4.2.1

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

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} + 3 x^{\frac{1}{4} \cdot 2} \cdot - 6 + 3 x^{\frac{1}{4}} {(- 6)}^{2} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.3.4.2.2.1

Factor $2$ out of $4$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} + 3 x^{\frac{1}{2 (2)} \cdot 2} \cdot - 6 + 3 x^{\frac{1}{4}} {(- 6)}^{2} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.2.2.2

Cancel the common factor.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} + 3 x^{\frac{1}{2 \cdot 2} \cdot 2} \cdot - 6 + 3 x^{\frac{1}{4}} {(- 6)}^{2} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.2.2.3

Rewrite the expression.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} + 3 x^{\frac{1}{2}} \cdot - 6 + 3 x^{\frac{1}{4}} {(- 6)}^{2} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.3

Multiply $- 6$ by $3$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} - 18 x^{\frac{1}{2}} + 3 x^{\frac{1}{4}} {(- 6)}^{2} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.4

Raise $- 6$ to the power of $2$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} - 18 x^{\frac{1}{2}} + 3 x^{\frac{1}{4}} \cdot 36 + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.5

Multiply $36$ by $3$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} - 18 x^{\frac{1}{2}} + 108 x^{\frac{1}{4}} + {(- 6)}^{3}) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.4.6

Raise $- 6$ to the power of $3$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 (x^{\frac{3}{4}} - 18 x^{\frac{1}{2}} + 108 x^{\frac{1}{4}} - 216) + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.5

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} + 24 (- 18 x^{\frac{1}{2}}) + 24 (108 x^{\frac{1}{4}}) + 24 \cdot - 216 + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.6

Simplify.

Tap for more steps...

Step 4.2.3.6.1

Multiply $- 18$ by $24$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 24 (108 x^{\frac{1}{4}}) + 24 \cdot - 216 + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.6.2

Multiply $108$ by $24$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} + 24 \cdot - 216 + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.6.3

Multiply $24$ by $- 216$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 {(x^{\frac{1}{4}} - 6)}^{2} + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.7

Rewrite ${(x^{\frac{1}{4}} - 6)}^{2}$ as $(x^{\frac{1}{4}} - 6) (x^{\frac{1}{4}} - 6)$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 ((x^{\frac{1}{4}} - 6) (x^{\frac{1}{4}} - 6)) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.8

Expand $(x^{\frac{1}{4}} - 6) (x^{\frac{1}{4}} - 6)$ using the FOIL Method.

Tap for more steps...

Step 4.2.3.8.1

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1}{4}} (x^{\frac{1}{4}} - 6) - 6 (x^{\frac{1}{4}} - 6)) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.8.2

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1}{4}} x^{\frac{1}{4}} + x^{\frac{1}{4}} \cdot - 6 - 6 (x^{\frac{1}{4}} - 6)) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.8.3

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1}{4}} x^{\frac{1}{4}} + x^{\frac{1}{4}} \cdot - 6 - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9

Simplify and combine like terms.

Tap for more steps...

Step 4.2.3.9.1

Simplify each term.

Tap for more steps...

Step 4.2.3.9.1.1

Multiply $x^{\frac{1}{4}}$ by $x^{\frac{1}{4}}$ by adding the exponents.

Tap for more steps...

Step 4.2.3.9.1.1.1

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

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1}{4} + \frac{1}{4}} + x^{\frac{1}{4}} \cdot - 6 - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.1.1.2

Combine the numerators over the common denominator.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1 + 1}{4}} + x^{\frac{1}{4}} \cdot - 6 - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.1.1.3

Add $1$ and $1$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{2}{4}} + x^{\frac{1}{4}} \cdot - 6 - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.1.1.4

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 4.2.3.9.1.1.4.1

Factor $2$ out of $2$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{2 (1)}{4}} + x^{\frac{1}{4}} \cdot - 6 - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.1.1.4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.3.9.1.1.4.2.1

Factor $2$ out of $4$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{2 \cdot 1}{2 \cdot 2}} + x^{\frac{1}{4}} \cdot - 6 - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.1.1.4.2.2

Cancel the common factor.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{2 \cdot 1}{2 \cdot 2}} + x^{\frac{1}{4}} \cdot - 6 - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.1.1.4.2.3

Rewrite the expression.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot - 6 - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.1.2

Move $- 6$ to the left of $x^{\frac{1}{4}}$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1}{2}} - 6 \cdot x^{\frac{1}{4}} - 6 x^{\frac{1}{4}} - 6 \cdot - 6) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.1.3

Multiply $- 6$ by $- 6$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1}{2}} - 6 x^{\frac{1}{4}} - 6 x^{\frac{1}{4}} + 36) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.9.2

Subtract $6 x^{\frac{1}{4}}$ from $- 6 x^{\frac{1}{4}}$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 (x^{\frac{1}{2}} - 12 x^{\frac{1}{4}} + 36) + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.10

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 x^{\frac{1}{2}} + 216 (- 12 x^{\frac{1}{4}}) + 216 \cdot 36 + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.11

Simplify.

Tap for more steps...

Step 4.2.3.11.1

Multiply $- 12$ by $216$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 x^{\frac{1}{2}} - 2592 x^{\frac{1}{4}} + 216 \cdot 36 + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.11.2

Multiply $216$ by $36$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 x^{\frac{1}{2}} - 2592 x^{\frac{1}{4}} + 7776 + 864 (x^{\frac{1}{4}} - 6) + 1296$

Step 4.2.3.12

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 x^{\frac{1}{2}} - 2592 x^{\frac{1}{4}} + 7776 + 864 x^{\frac{1}{4}} + 864 \cdot - 6 + 1296$

Step 4.2.3.13

Multiply $864$ by $- 6$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 x^{\frac{1}{2}} - 2592 x^{\frac{1}{4}} + 7776 + 864 x^{\frac{1}{4}} - 5184 + 1296$

Step 4.2.4

Simplify by adding terms.

Tap for more steps...

Step 4.2.4.1

Combine the opposite terms in $x - 24 x^{\frac{3}{4}} + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 24 x^{\frac{3}{4}} - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 x^{\frac{1}{2}} - 2592 x^{\frac{1}{4}} + 7776 + 864 x^{\frac{1}{4}} - 5184 + 1296$ .

Tap for more steps...

Step 4.2.4.1.1

Add $- 24 x^{\frac{3}{4}}$ and $24 x^{\frac{3}{4}}$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 + 0 - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 x^{\frac{1}{2}} - 2592 x^{\frac{1}{4}} + 7776 + 864 x^{\frac{1}{4}} - 5184 + 1296$

Step 4.2.4.1.2

Add $x + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296$ and $0$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 - 432 x^{\frac{1}{2}} + 2592 x^{\frac{1}{4}} - 5184 + 216 x^{\frac{1}{2}} - 2592 x^{\frac{1}{4}} + 7776 + 864 x^{\frac{1}{4}} - 5184 + 1296$

Step 4.2.4.1.3

Subtract $2592 x^{\frac{1}{4}}$ from $2592 x^{\frac{1}{4}}$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 - 432 x^{\frac{1}{2}} - 5184 + 216 x^{\frac{1}{2}} + 0 + 7776 + 864 x^{\frac{1}{4}} - 5184 + 1296$

Step 4.2.4.1.4

Add $x + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 - 432 x^{\frac{1}{2}} - 5184 + 216 x^{\frac{1}{2}}$ and $0$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 216 x^{\frac{1}{2}} - 864 x^{\frac{1}{4}} + 1296 - 432 x^{\frac{1}{2}} - 5184 + 216 x^{\frac{1}{2}} + 7776 + 864 x^{\frac{1}{4}} - 5184 + 1296$

Step 4.2.4.1.5

Add $- 864 x^{\frac{1}{4}}$ and $864 x^{\frac{1}{4}}$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 216 x^{\frac{1}{2}} + 1296 - 432 x^{\frac{1}{2}} - 5184 + 216 x^{\frac{1}{2}} + 7776 + 0 - 5184 + 1296$

Step 4.2.4.1.6

Add $x + 216 x^{\frac{1}{2}} + 1296 - 432 x^{\frac{1}{2}} - 5184 + 216 x^{\frac{1}{2}} + 7776$ and $0$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 216 x^{\frac{1}{2}} + 1296 - 432 x^{\frac{1}{2}} - 5184 + 216 x^{\frac{1}{2}} + 7776 - 5184 + 1296$

Step 4.2.4.2

Subtract $432 x^{\frac{1}{2}}$ from $216 x^{\frac{1}{2}}$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 216 x^{\frac{1}{2}} + 1296 - 5184 + 216 x^{\frac{1}{2}} + 7776 - 5184 + 1296$

Step 4.2.4.3

Combine the opposite terms in $x - 216 x^{\frac{1}{2}} + 1296 - 5184 + 216 x^{\frac{1}{2}} + 7776 - 5184 + 1296$ .

Tap for more steps...

Step 4.2.4.3.1

Add $- 216 x^{\frac{1}{2}}$ and $216 x^{\frac{1}{2}}$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 0 + 1296 - 5184 + 7776 - 5184 + 1296$

Step 4.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 1296 - 5184 + 7776 - 5184 + 1296$

Step 4.2.4.4

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.4.4.1

Subtract $5184$ from $1296$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 3888 + 7776 - 5184 + 1296$

Step 4.2.4.4.2

Add $- 3888$ and $7776$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 3888 - 5184 + 1296$

Step 4.2.4.4.3

Subtract $5184$ from $3888$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x - 1296 + 1296$

Step 4.2.4.5

Combine the opposite terms in $x - 1296 + 1296$ .

Tap for more steps...

Step 4.2.4.5.1

Add $- 1296$ and $1296$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x + 0$

Step 4.2.4.5.2

Add $x$ and $0$ .

$f^{- 1} (x^{\frac{1}{4}} - 6) = x$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296) = {(x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296)}^{\frac{1}{4}} - 6$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296$ is the inverse of $f (x) = x^{\frac{1}{4}} - 6$ .

$f^{- 1} (x) = x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296$