Solve for y (5y-1)^(-1/3)-6(5y-1)^(-4/3)=0

${(5 y - 1)}^{- \frac{1}{3}} - 6 {(5 y - 1)}^{- \frac{4}{3}} = 0$

Step 1

Simplify each term.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} - 6 {(5 y - 1)}^{- \frac{4}{3}} = 0$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} - 6 \frac{1}{{(5 y - 1)}^{\frac{4}{3}}} = 0$

Combine $- 6$ and $\frac{1}{{(5 y - 1)}^{\frac{4}{3}}}$ .

$\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} + \frac{- 6}{{(5 y - 1)}^{\frac{4}{3}}} = 0$

Move the negative in front of the fraction.

$\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} = 0$

Step 2

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

${(5 y - 1)}^{\frac{1}{3}}, {(5 y - 1)}^{\frac{4}{3}}, 1$

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $5 y - 1$ is $5 y - 1$ itself.

$(5 y - 1) = 5 y - 1$

$(5 y - 1)$ occurs $1$ time.

The factors for $5 y - 1$ are $(5 y - 1) \cdot (5 y - 1)$ , which is $5 y - 1$ multiplied by itself $2$ times.

$(5 y - 1) = (5 y - 1) \cdot (5 y - 1)$

$(5 y - 1)$ occurs $2$ times.

The LCM of ${(5 y - 1)}^{\frac{1}{3}}, {(5 y - 1)}^{\frac{4}{3}}$ is the result of multiplying all factors the greatest number of times they occur in either term.

${(5 y - 1)}^{\frac{4}{3}}$

Step 3

Multiply each term in $\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} = 0$ by ${(5 y - 1)}^{\frac{4}{3}}$ to eliminate the fractions.

Multiply each term in $\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} = 0$ by ${(5 y - 1)}^{\frac{4}{3}}$ .

$\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} {(5 y - 1)}^{\frac{4}{3}} - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} {(5 y - 1)}^{\frac{4}{3}} = 0 {(5 y - 1)}^{\frac{4}{3}}$

Simplify the left side.

Simplify each term.

Cancel the common factor of ${(5 y - 1)}^{\frac{1}{3}}$ .

Factor ${(5 y - 1)}^{\frac{1}{3}}$ out of ${(5 y - 1)}^{\frac{4}{3}}$ .

$\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} ({(5 y - 1)}^{\frac{1}{3}} {(5 y - 1)}^{\frac{3}{3}}) - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} {(5 y - 1)}^{\frac{4}{3}} = 0 {(5 y - 1)}^{\frac{4}{3}}$

Cancel the common factor.

$\frac{1}{{(5 y - 1)}^{\frac{1}{3}}} ({(5 y - 1)}^{\frac{1}{3}} {(5 y - 1)}^{\frac{3}{3}}) - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} {(5 y - 1)}^{\frac{4}{3}} = 0 {(5 y - 1)}^{\frac{4}{3}}$

Rewrite the expression.

${(5 y - 1)}^{\frac{3}{3}} - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} {(5 y - 1)}^{\frac{4}{3}} = 0 {(5 y - 1)}^{\frac{4}{3}}$

Divide $3$ by $3$ .

${(5 y - 1)}^{1} - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} {(5 y - 1)}^{\frac{4}{3}} = 0 {(5 y - 1)}^{\frac{4}{3}}$

Simplify.

$5 y - 1 - \frac{6}{{(5 y - 1)}^{\frac{4}{3}}} {(5 y - 1)}^{\frac{4}{3}} = 0 {(5 y - 1)}^{\frac{4}{3}}$

Cancel the common factor of ${(5 y - 1)}^{\frac{4}{3}}$ .

Move the leading negative in $- \frac{6}{{(5 y - 1)}^{\frac{4}{3}}}$ into the numerator.

$5 y - 1 + \frac{- 6}{{(5 y - 1)}^{\frac{4}{3}}} {(5 y - 1)}^{\frac{4}{3}} = 0 {(5 y - 1)}^{\frac{4}{3}}$

Cancel the common factor.

$5 y - 1 + \frac{- 6}{{(5 y - 1)}^{\frac{4}{3}}} {(5 y - 1)}^{\frac{4}{3}} = 0 {(5 y - 1)}^{\frac{4}{3}}$

Rewrite the expression.

$5 y - 1 - 6 = 0 {(5 y - 1)}^{\frac{4}{3}}$

Subtract $6$ from $- 1$ .

$5 y - 7 = 0 {(5 y - 1)}^{\frac{4}{3}}$

Simplify the right side.

Multiply $0$ by ${(5 y - 1)}^{\frac{4}{3}}$ .

$5 y - 7 = 0$

Step 4

Solve the equation.

Add $7$ to both sides of the equation.

$5 y = 7$

Divide each term in $5 y = 7$ by $5$ and simplify.

Divide each term in $5 y = 7$ by $5$ .

$\frac{5 y}{5} = \frac{7}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{7}{5}$

Divide $y$ by $1$ .

$y = \frac{7}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{7}{5}$

Decimal Form:

$y = 1.4$

Mixed Number Form:

$y = 1 \frac{2}{5}$

Solve for y (5y-1)^(-1/3)-6(5y-1)^(-4/3)=0

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