Solve for y (2y-3)^(2/3)=(3y)^(1/3)

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

Step 1

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

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

Step 2

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

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

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

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 3$ by $2$ .

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

Multiply $2$ by $- 3$ .

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

Multiply $- 3$ by $- 3$ .

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

Subtract $6 y$ from $- 6 y$ .

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

Subtract $3 y$ from $- 12 y$ .

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

Step 5

Factor by grouping.

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 $b = - 15$ .

Factor $- 15$ out of $- 15 y$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 6

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

$4 y - 3 = 0$

$y - 3 = 0$

Step 7

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

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

$4 y - 3 = 0$

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

Add $3$ to both sides of the equation.

$4 y = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 8

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

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

$y - 3 = 0$

Add $3$ to both sides of the equation.

$y = 3$

Step 9

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

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = 0.75, 3$

Solve for y (2y-3)^(2/3)=(3y)^(1/3)

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