Finite Math Examples

$5 z^{\frac{13}{5}} - 31 z^{\frac{8}{5}} + 6 z^{\frac{3}{5}} = 0$

Step 1

Factor $z^{\frac{3}{5}}$ out of $5 z^{\frac{13}{5}} - 31 z^{\frac{8}{5}} + 6 z^{\frac{3}{5}}$ .

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Step 1.1

Factor $z^{\frac{3}{5}}$ out of $5 z^{\frac{13}{5}}$ .

$z^{\frac{3}{5}} (5 z^{\frac{10}{5}}) - 31 z^{\frac{8}{5}} + 6 z^{\frac{3}{5}} = 0$

Step 1.2

Factor $z^{\frac{3}{5}}$ out of $- 31 z^{\frac{8}{5}}$ .

$z^{\frac{3}{5}} (5 z^{\frac{10}{5}}) + z^{\frac{3}{5}} (- 31 z^{\frac{5}{5}}) + 6 z^{\frac{3}{5}} = 0$

Step 1.3

Factor $z^{\frac{3}{5}}$ out of $6 z^{\frac{3}{5}}$ .

$z^{\frac{3}{5}} (5 z^{\frac{10}{5}}) + z^{\frac{3}{5}} (- 31 z^{\frac{5}{5}}) + z^{\frac{3}{5}} \cdot 6 = 0$

Step 1.4

Factor $z^{\frac{3}{5}}$ out of $z^{\frac{3}{5}} (5 z^{\frac{10}{5}}) + z^{\frac{3}{5}} (- 31 z^{\frac{5}{5}})$ .

$z^{\frac{3}{5}} (5 z^{\frac{10}{5}} - 31 z^{\frac{5}{5}}) + z^{\frac{3}{5}} \cdot 6 = 0$

Step 1.5

Factor $z^{\frac{3}{5}}$ out of $z^{\frac{3}{5}} (5 z^{\frac{10}{5}} - 31 z^{\frac{5}{5}}) + z^{\frac{3}{5}} \cdot 6$ .

$z^{\frac{3}{5}} (5 z^{\frac{10}{5}} - 31 z^{\frac{5}{5}} + 6) = 0$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 5 \cdot 6 = 30$ and whose sum is $b = - 31$ .

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Step 2.1.1

Factor $- 31$ out of $- 31 z^{\frac{5}{5}}$ .

$z^{\frac{3}{5}} (5 z^{\frac{10}{5}} - 31 z^{\frac{5}{5}} + 6) = 0$

Step 2.1.2

Rewrite $- 31$ as $- 1$ plus $- 30$

$z^{\frac{3}{5}} (5 z^{\frac{10}{5}} + (- 1 - 30) z^{\frac{5}{5}} + 6) = 0$

Step 2.1.3

Apply the distributive property.

$z^{\frac{3}{5}} (5 z^{\frac{10}{5}} - 1 z^{\frac{5}{5}} - 30 z^{\frac{5}{5}} + 6) = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$z^{\frac{3}{5}} ((5 z^{\frac{10}{5}} - 1 z^{\frac{5}{5}}) - 30 z^{\frac{5}{5}} + 6) = 0$

Step 2.2.2

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

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

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $5 z^{\frac{5}{5}} - 1$ .

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

Step 3

Divide $5$ by $5$ .

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

Step 4

Divide $5$ by $5$ .

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

Step 5

Remove unnecessary parentheses.

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

Step 6

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

$z^{\frac{3}{5}} = 0$

$5 z - 1 = 0$

$z - 6 = 0$

Step 7

Set $z^{\frac{3}{5}}$ equal to $0$ and solve for $z$ .

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Step 7.1

Set $z^{\frac{3}{5}}$ equal to $0$ .

$z^{\frac{3}{5}} = 0$

Step 7.2

Solve $z^{\frac{3}{5}} = 0$ for $z$ .

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Step 7.2.1

Raise each side of the equation to the power of $\frac{5}{3}$ to eliminate the fractional exponent on the left side.

${(z^{\frac{3}{5}})}^{\frac{5}{3}} = 0^{\frac{5}{3}}$

Step 7.2.2

Simplify the exponent.

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Step 7.2.2.1

Simplify the left side.

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Step 7.2.2.1.1

Simplify ${(z^{\frac{3}{5}})}^{\frac{5}{3}}$ .

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Step 7.2.2.1.1.1

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

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Step 7.2.2.1.1.1.1

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

$z^{\frac{3}{5} \cdot \frac{5}{3}} = 0^{\frac{5}{3}}$

Step 7.2.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 7.2.2.1.1.1.2.1

Cancel the common factor.

$z^{\frac{3}{5} \cdot \frac{5}{3}} = 0^{\frac{5}{3}}$

Step 7.2.2.1.1.1.2.2

Rewrite the expression.

$z^{\frac{1}{5} \cdot 5} = 0^{\frac{5}{3}}$

Step 7.2.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 7.2.2.1.1.1.3.1

Cancel the common factor.

$z^{\frac{1}{5} \cdot 5} = 0^{\frac{5}{3}}$

Step 7.2.2.1.1.1.3.2

Rewrite the expression.

$z^{1} = 0^{\frac{5}{3}}$

Step 7.2.2.1.1.2

Simplify.

$z = 0^{\frac{5}{3}}$

Step 7.2.2.2

Simplify the right side.

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Step 7.2.2.2.1

Simplify $0^{\frac{5}{3}}$ .

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Step 7.2.2.2.1.1

Simplify the expression.

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Step 7.2.2.2.1.1.1

Rewrite $0$ as $0^{3}$ .

$z = {(0^{3})}^{\frac{5}{3}}$

Step 7.2.2.2.1.1.2

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

$z = 0^{3 (\frac{5}{3})}$

Step 7.2.2.2.1.2

Cancel the common factor of $3$ .

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Step 7.2.2.2.1.2.1

Cancel the common factor.

$z = 0^{3 (\frac{5}{3})}$

Step 7.2.2.2.1.2.2

Rewrite the expression.

$z = 0^{5}$

Step 7.2.2.2.1.3

Raising $0$ to any positive power yields $0$ .

$z = 0$

Step 8

Set $5 z - 1$ equal to $0$ and solve for $z$ .

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Step 8.1

Set $5 z - 1$ equal to $0$ .

$5 z - 1 = 0$

Step 8.2

Solve $5 z - 1 = 0$ for $z$ .

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Step 8.2.1

Add $1$ to both sides of the equation.

$5 z = 1$

Step 8.2.2

Divide each term in $5 z = 1$ by $5$ and simplify.

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Step 8.2.2.1

Divide each term in $5 z = 1$ by $5$ .

$\frac{5 z}{5} = \frac{1}{5}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $5$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{5 z}{5} = \frac{1}{5}$

Step 8.2.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{1}{5}$

Step 9

Set $z - 6$ equal to $0$ and solve for $z$ .

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Step 9.1

Set $z - 6$ equal to $0$ .

$z - 6 = 0$

Step 9.2

Add $6$ to both sides of the equation.

$z = 6$

Step 10

The final solution is all the values that make $z^{\frac{3}{5}} (5 z - 1) (z - 6) = 0$ true.

$z = 0, \frac{1}{5}, 6$