Precalculus Examples

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

Step 1

Eliminate the fractional exponents by multiplying both exponents by the LCD.

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

Step 2

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

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Step 2.1

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

${(2 x - 3)}^{\frac{2}{5} \cdot 5} = {({(3 x)}^{\frac{1}{5}})}^{5}$

Step 2.2

Cancel the common factor of $5$ .

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Step 2.2.1

Cancel the common factor.

${(2 x - 3)}^{\frac{2}{5} \cdot 5} = {({(3 x)}^{\frac{1}{5}})}^{5}$

Step 2.2.2

Rewrite the expression.

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

Step 3

Simplify ${({(3 x)}^{\frac{1}{5}})}^{5}$ .

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Step 3.1

Multiply the exponents in ${({(3 x)}^{\frac{1}{5}})}^{5}$ .

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Step 3.1.1

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

${(2 x - 3)}^{2} = {(3 x)}^{\frac{1}{5} \cdot 5}$

Step 3.1.2

Cancel the common factor of $5$ .

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Step 3.1.2.1

Cancel the common factor.

${(2 x - 3)}^{2} = {(3 x)}^{\frac{1}{5} \cdot 5}$

Step 3.1.2.2

Rewrite the expression.

${(2 x - 3)}^{2} = {(3 x)}^{1}$

Step 3.2

Simplify.

${(2 x - 3)}^{2} = 3 x$

Step 4

Move all terms containing $x$ to the left side of the equation.

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Step 4.1

Subtract $3 x$ from both sides of the equation.

${(2 x - 3)}^{2} - 3 x = 0$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite ${(2 x - 3)}^{2}$ as $(2 x - 3) (2 x - 3)$ .

$(2 x - 3) (2 x - 3) - 3 x = 0$

Step 4.2.2

Expand $(2 x - 3) (2 x - 3)$ using the FOIL Method.

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Step 4.2.2.1

Apply the distributive property.

$2 x (2 x - 3) - 3 (2 x - 3) - 3 x = 0$

Step 4.2.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 3 - 3 (2 x - 3) - 3 x = 0$

Step 4.2.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 3 x = 0$

Step 4.2.3

Simplify and combine like terms.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 3 x = 0$

Step 4.2.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - 3 x = 0$

Step 4.2.3.1.2.2

Multiply $x$ by $x$ .

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

Step 4.2.3.1.3

Multiply $2$ by $2$ .

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

Step 4.2.3.1.4

Multiply $- 3$ by $2$ .

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

Step 4.2.3.1.5

Multiply $2$ by $- 3$ .

$4 x^{2} - 6 x - 6 x - 3 \cdot - 3 - 3 x = 0$

Step 4.2.3.1.6

Multiply $- 3$ by $- 3$ .

$4 x^{2} - 6 x - 6 x + 9 - 3 x = 0$

Step 4.2.3.2

Subtract $6 x$ from $- 6 x$ .

$4 x^{2} - 12 x + 9 - 3 x = 0$

Step 4.3

Subtract $3 x$ from $- 12 x$ .

$4 x^{2} - 15 x + 9 = 0$

Step 5

Factor by grouping.

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Step 5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 9 = 36$ and whose sum is $b = - 15$ .

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Step 5.1.1

Factor $- 15$ out of $- 15 x$ .

$4 x^{2} - 15 x + 9 = 0$

Step 5.1.2

Rewrite $- 15$ as $- 3$ plus $- 12$

$4 x^{2} + (- 3 - 12) x + 9 = 0$

Step 5.1.3

Apply the distributive property.

$4 x^{2} - 3 x - 12 x + 9 = 0$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$(4 x^{2} - 3 x) - 12 x + 9 = 0$

Step 5.2.2

Factor out the greatest common factor (GCF) from each group.

$x (4 x - 3) - 3 (4 x - 3) = 0$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 3$ .

$(4 x - 3) (x - 3) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x - 3 = 0$

$x - 3 = 0$

Step 7

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 7.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 7.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 7.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 7.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 8

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 8.2

Add $3$ to both sides of the equation.

$x = 3$

Step 9

The final solution is all the values that make $(4 x - 3) (x - 3) = 0$ true.

$x = \frac{3}{4}, 3$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{4}, 3$

Decimal Form:

$x = 0.75, 3$