Calculus Examples

$2 {(x - 2)}^{- \frac{1}{3}} - \frac{2}{3} \cdot (x {(x - 2)}^{- \frac{4}{3}}) = 0$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 \frac{1}{{(x - 2)}^{\frac{1}{3}}} - \frac{2}{3} \cdot (x {(x - 2)}^{- \frac{4}{3}}) = 0$

Step 1.2

Combine $2$ and $\frac{1}{{(x - 2)}^{\frac{1}{3}}}$ .

$\frac{2}{{(x - 2)}^{\frac{1}{3}}} - \frac{2}{3} \cdot (x {(x - 2)}^{- \frac{4}{3}}) = 0$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{{(x - 2)}^{\frac{1}{3}}} - \frac{2}{3} \cdot (x \frac{1}{{(x - 2)}^{\frac{4}{3}}}) = 0$

Step 1.4

Combine $x$ and $\frac{1}{{(x - 2)}^{\frac{4}{3}}}$ .

$\frac{2}{{(x - 2)}^{\frac{1}{3}}} - \frac{2}{3} \cdot \frac{x}{{(x - 2)}^{\frac{4}{3}}} = 0$

Step 1.5

Multiply $\frac{x}{{(x - 2)}^{\frac{4}{3}}}$ by $\frac{2}{3}$ .

$\frac{2}{{(x - 2)}^{\frac{1}{3}}} - \frac{x \cdot 2}{{(x - 2)}^{\frac{4}{3}} \cdot 3} = 0$

Step 1.6

Move $2$ to the left of $x$ .

$\frac{2}{{(x - 2)}^{\frac{1}{3}}} - \frac{2 \cdot x}{{(x - 2)}^{\frac{4}{3}} \cdot 3} = 0$

Step 1.7

Move $3$ to the left of ${(x - 2)}^{\frac{4}{3}}$ .

$\frac{2}{{(x - 2)}^{\frac{1}{3}}} - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} = 0$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

${(x - 2)}^{\frac{1}{3}}, 3 {(x - 2)}^{\frac{4}{3}}, 1$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.6

The LCM of $1, 3, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 2.7

The factor for $x - 2$ is $x - 2$ itself.

$(x - 2) = x - 2$

$(x - 2)$ occurs $1$ time.

Step 2.8

The factors for $x - 2$ are $(x - 2) \cdot (x - 2)$ , which is $x - 2$ multiplied by itself $2$ times.

$(x - 2) = (x - 2) \cdot (x - 2)$

$(x - 2)$ occurs $2$ times.

Step 2.9

The LCM of ${(x - 2)}^{\frac{1}{3}}, {(x - 2)}^{\frac{4}{3}}$ is the result of multiplying all factors the greatest number of times they occur in either term.

${(x - 2)}^{\frac{4}{3}}$

Step 2.10

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$3 {(x - 2)}^{\frac{4}{3}}$

Step 3

Multiply each term in $\frac{2}{{(x - 2)}^{\frac{1}{3}}} - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} = 0$ by $3 {(x - 2)}^{\frac{4}{3}}$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{2}{{(x - 2)}^{\frac{1}{3}}} - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} = 0$ by $3 {(x - 2)}^{\frac{4}{3}}$ .

$\frac{2}{{(x - 2)}^{\frac{1}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Rewrite using the commutative property of multiplication.

$3 \frac{2}{{(x - 2)}^{\frac{1}{3}}} {(x - 2)}^{\frac{4}{3}} - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.2

Multiply $3 \frac{2}{{(x - 2)}^{\frac{1}{3}}}$ .

Tap for more steps...

Step 3.2.1.2.1

Combine $3$ and $\frac{2}{{(x - 2)}^{\frac{1}{3}}}$ .

$\frac{3 \cdot 2}{{(x - 2)}^{\frac{1}{3}}} {(x - 2)}^{\frac{4}{3}} - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.2.2

Multiply $3$ by $2$ .

$\frac{6}{{(x - 2)}^{\frac{1}{3}}} {(x - 2)}^{\frac{4}{3}} - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.3

Cancel the common factor of ${(x - 2)}^{\frac{1}{3}}$ .

Tap for more steps...

Step 3.2.1.3.1

Factor ${(x - 2)}^{\frac{1}{3}}$ out of ${(x - 2)}^{\frac{4}{3}}$ .

$\frac{6}{{(x - 2)}^{\frac{1}{3}}} ({(x - 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{3}{3}}) - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.3.2

Cancel the common factor.

$\frac{6}{{(x - 2)}^{\frac{1}{3}}} ({(x - 2)}^{\frac{1}{3}} {(x - 2)}^{\frac{3}{3}}) - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.3.3

Rewrite the expression.

$6 {(x - 2)}^{\frac{3}{3}} - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.4

Divide $3$ by $3$ .

$6 {(x - 2)}^{1} - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.5

Simplify.

$6 (x - 2) - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.6

Apply the distributive property.

$6 x + 6 \cdot - 2 - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.7

Multiply $6$ by $- 2$ .

$6 x - 12 - \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.8

Cancel the common factor of $3 {(x - 2)}^{\frac{4}{3}}$ .

Tap for more steps...

Step 3.2.1.8.1

Move the leading negative in $- \frac{2 x}{3 {(x - 2)}^{\frac{4}{3}}}$ into the numerator.

$6 x - 12 + \frac{- 2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.8.2

Cancel the common factor.

$6 x - 12 + \frac{- 2 x}{3 {(x - 2)}^{\frac{4}{3}}} (3 {(x - 2)}^{\frac{4}{3}}) = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.1.8.3

Rewrite the expression.

$6 x - 12 - 2 x = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.2.2

Subtract $2 x$ from $6 x$ .

$4 x - 12 = 0 (3 {(x - 2)}^{\frac{4}{3}})$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply $0 (3 {(x - 2)}^{\frac{4}{3}})$ .

Tap for more steps...

Step 3.3.1.1

Multiply $3$ by $0$ .

$4 x - 12 = 0 {(x - 2)}^{\frac{4}{3}}$

Step 3.3.1.2

Multiply $0$ by ${(x - 2)}^{\frac{4}{3}}$ .

$4 x - 12 = 0$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Add $12$ to both sides of the equation.

$4 x = 12$

Step 4.2

Divide each term in $4 x = 12$ by $4$ and simplify.

Tap for more steps...

Step 4.2.1

Divide each term in $4 x = 12$ by $4$ .

$\frac{4 x}{4} = \frac{12}{4}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{12}{4}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{4}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Divide $12$ by $4$ .

$x = 3$