Solve for y (3y-2)^(2/5)=(8y)^(1/5)

${(3 y - 2)}^{\frac{2}{5}} = {(8 y)}^{\frac{1}{5}}$

Step 1

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

${({(3 y - 2)}^{\frac{2}{5}})}^{5} = {({(8 y)}^{\frac{1}{5}})}^{5}$

Step 2

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

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

${(3 y - 2)}^{\frac{2}{5} \cdot 5} = {({(8 y)}^{\frac{1}{5}})}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(3 y - 2)}^{\frac{2}{5} \cdot 5} = {({(8 y)}^{\frac{1}{5}})}^{5}$

Rewrite the expression.

${(3 y - 2)}^{2} = {({(8 y)}^{\frac{1}{5}})}^{5}$

Step 3

Simplify ${({(8 y)}^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${({(8 y)}^{\frac{1}{5}})}^{5}$ .

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

${(3 y - 2)}^{2} = {(8 y)}^{\frac{1}{5} \cdot 5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(3 y - 2)}^{2} = {(8 y)}^{\frac{1}{5} \cdot 5}$

Rewrite the expression.

${(3 y - 2)}^{2} = {(8 y)}^{1}$

Simplify.

${(3 y - 2)}^{2} = 8 y$

Step 4

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

Subtract $8 y$ from both sides of the equation.

${(3 y - 2)}^{2} - 8 y = 0$

Simplify each term.

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

$(3 y - 2) (3 y - 2) - 8 y = 0$

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

Apply the distributive property.

$3 y (3 y - 2) - 2 (3 y - 2) - 8 y = 0$

Apply the distributive property.

$3 y (3 y) + 3 y \cdot - 2 - 2 (3 y - 2) - 8 y = 0$

Apply the distributive property.

$3 y (3 y) + 3 y \cdot - 2 - 2 (3 y) - 2 \cdot - 2 - 8 y = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 \cdot 3 y \cdot y + 3 y \cdot - 2 - 2 (3 y) - 2 \cdot - 2 - 8 y = 0$

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

Move $y$ .

$3 \cdot 3 (y \cdot y) + 3 y \cdot - 2 - 2 (3 y) - 2 \cdot - 2 - 8 y = 0$

Multiply $y$ by $y$ .

$3 \cdot 3 y^{2} + 3 y \cdot - 2 - 2 (3 y) - 2 \cdot - 2 - 8 y = 0$

Multiply $3$ by $3$ .

$9 y^{2} + 3 y \cdot - 2 - 2 (3 y) - 2 \cdot - 2 - 8 y = 0$

Multiply $- 2$ by $3$ .

$9 y^{2} - 6 y - 2 (3 y) - 2 \cdot - 2 - 8 y = 0$

Multiply $3$ by $- 2$ .

$9 y^{2} - 6 y - 6 y - 2 \cdot - 2 - 8 y = 0$

Multiply $- 2$ by $- 2$ .

$9 y^{2} - 6 y - 6 y + 4 - 8 y = 0$

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

$9 y^{2} - 12 y + 4 - 8 y = 0$

Subtract $8 y$ from $- 12 y$ .

$9 y^{2} - 20 y + 4 = 0$

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 $a \cdot c = 9 \cdot 4 = 36$ and whose sum is $b = - 20$ .

Factor $- 20$ out of $- 20 y$ .

$9 y^{2} - 20 y + 4 = 0$

Rewrite $- 20$ as $- 2$ plus $- 18$

$9 y^{2} + (- 2 - 18) y + 4 = 0$

Apply the distributive property.

$9 y^{2} - 2 y - 18 y + 4 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 y^{2} - 2 y) - 18 y + 4 = 0$

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

$y (9 y - 2) - 2 (9 y - 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $9 y - 2$ .

$(9 y - 2) (y - 2) = 0$

Step 6

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

$9 y - 2 = 0$

$y - 2 = 0$

Step 7

Set $9 y - 2$ equal to $0$ and solve for $y$ .

Set $9 y - 2$ equal to $0$ .

$9 y - 2 = 0$

Solve $9 y - 2 = 0$ for $y$ .

Add $2$ to both sides of the equation.

$9 y = 2$

Divide each term in $9 y = 2$ by $9$ and simplify.

Divide each term in $9 y = 2$ by $9$ .

$\frac{9 y}{9} = \frac{2}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} = \frac{2}{9}$

Divide $y$ by $1$ .

$y = \frac{2}{9}$

Step 8

Set $y - 2$ equal to $0$ and solve for $y$ .

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Add $2$ to both sides of the equation.

$y = 2$

Step 9

The final solution is all the values that make $(9 y - 2) (y - 2) = 0$ true.

$y = \frac{2}{9}, 2$

Step 10

The result can be shown in multiple forms.

Exact Form:

$y = \frac{2}{9}, 2$

Decimal Form:

$y = 0.\overline{2}, 2$

Solve for y (3y-2)^(2/5)=(8y)^(1/5)

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