Solve for x (5x-4)^(2/3)=x^(1/3)

${(5 x - 4)}^{\frac{2}{3}} = x^{\frac{1}{3}}$

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

${({(5 x - 4)}^{\frac{2}{3}})}^{3} = {(x^{\frac{1}{3}})}^{3}$

Multiply the exponents in ${({(5 x - 4)}^{\frac{2}{3}})}^{3}$ .

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

${(5 x - 4)}^{\frac{2}{3} \cdot 3} = {(x^{\frac{1}{3}})}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(5 x - 4)}^{\frac{2}{3} \cdot 3} = {(x^{\frac{1}{3}})}^{3}$

Rewrite the expression.

${(5 x - 4)}^{2} = {(x^{\frac{1}{3}})}^{3}$

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

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

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

${(5 x - 4)}^{2} = x^{\frac{1}{3} \cdot 3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(5 x - 4)}^{2} = x^{\frac{1}{3} \cdot 3}$

Rewrite the expression.

${(5 x - 4)}^{2} = x^{1}$

Simplify.

${(5 x - 4)}^{2} = x$

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

Subtract $x$ from both sides of the equation.

${(5 x - 4)}^{2} - x = 0$

Simplify each term.

Rewrite ${(5 x - 4)}^{2}$ as $(5 x - 4) (5 x - 4)$ .

$(5 x - 4) (5 x - 4) - x = 0$

Expand $(5 x - 4) (5 x - 4)$ using the FOIL Method.

Apply the distributive property.

$5 x (5 x - 4) - 4 (5 x - 4) - x = 0$

Apply the distributive property.

$5 x (5 x) + 5 x \cdot - 4 - 4 (5 x - 4) - x = 0$

Apply the distributive property.

$5 x (5 x) + 5 x \cdot - 4 - 4 (5 x) - 4 \cdot - 4 - x = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$5 \cdot 5 x \cdot x + 5 x \cdot - 4 - 4 (5 x) - 4 \cdot - 4 - x = 0$

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

Move $x$ .

$5 \cdot 5 (x \cdot x) + 5 x \cdot - 4 - 4 (5 x) - 4 \cdot - 4 - x = 0$

Multiply $x$ by $x$ .

$5 \cdot 5 x^{2} + 5 x \cdot - 4 - 4 (5 x) - 4 \cdot - 4 - x = 0$

Multiply $5$ by $5$ .

$25 x^{2} + 5 x \cdot - 4 - 4 (5 x) - 4 \cdot - 4 - x = 0$

Multiply $- 4$ by $5$ .

$25 x^{2} - 20 x - 4 (5 x) - 4 \cdot - 4 - x = 0$

Multiply $5$ by $- 4$ .

$25 x^{2} - 20 x - 20 x - 4 \cdot - 4 - x = 0$

Multiply $- 4$ by $- 4$ .

$25 x^{2} - 20 x - 20 x + 16 - x = 0$

Subtract $20 x$ from $- 20 x$ .

$25 x^{2} - 40 x + 16 - x = 0$

Subtract $x$ from $- 40 x$ .

$25 x^{2} - 41 x + 16 = 0$

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 $a \cdot c = 25 \cdot 16 = 400$ and whose sum is $b = - 41$ .

Factor $- 41$ out of $- 41 x$ .

$25 x^{2} - 41 x + 16 = 0$

Rewrite $- 41$ as $- 16$ plus $- 25$

$25 x^{2} + (- 16 - 25) x + 16 = 0$

Apply the distributive property.

$25 x^{2} - 16 x - 25 x + 16 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(25 x^{2} - 16 x) - 25 x + 16 = 0$

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

$x (25 x - 16) - (25 x - 16) = 0$

Factor the polynomial by factoring out the greatest common factor, $25 x - 16$ .

$(25 x - 16) (x - 1) = 0$

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

$25 x - 16 = 0$

$x - 1 = 0$

Set $25 x - 16$ equal to $0$ and solve for $x$ .

Set $25 x - 16$ equal to $0$ .

$25 x - 16 = 0$

Solve $25 x - 16 = 0$ for $x$ .

Add $16$ to both sides of the equation.

$25 x = 16$

Divide each term in $25 x = 16$ by $25$ and simplify.

Divide each term in $25 x = 16$ by $25$ .

$\frac{25 x}{25} = \frac{16}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 x}{25} = \frac{16}{25}$

Divide $x$ by $1$ .

$x = \frac{16}{25}$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(25 x - 16) (x - 1) = 0$ true.

$x = \frac{16}{25}, 1$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{16}{25}, 1$

Decimal Form:

$x = 0.64, 1$

Solve for x (5x-4)^(2/3)=x^(1/3)

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