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

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

Step 1

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

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

Step 2

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

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

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

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

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

Multiply $z$ by $z$ .

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

Multiply $3$ by $3$ .

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

Multiply $- 2$ by $3$ .

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

Multiply $3$ by $- 2$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $6 z$ from $- 6 z$ .

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

Subtract $8 z$ from $- 12 z$ .

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

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 $- 20 z$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 6

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

$9 z - 2 = 0$

$z - 2 = 0$

Step 7

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

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

$9 z - 2 = 0$

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

Add $2$ to both sides of the equation.

$9 z = 2$

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

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

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

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

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

Divide $z$ by $1$ .

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

Step 8

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

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

$z - 2 = 0$

Add $2$ to both sides of the equation.

$z = 2$

Step 9

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

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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

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

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