Precalculus Examples

$3 z^{\frac{14}{5}} - 19 z^{\frac{9}{5}} + 28 z^{\frac{4}{5}} = 0$

Step 1

Factor $z^{\frac{4}{5}}$ out of $3 z^{\frac{14}{5}} - 19 z^{\frac{9}{5}} + 28 z^{\frac{4}{5}}$ .

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Step 1.1

Factor $z^{\frac{4}{5}}$ out of $3 z^{\frac{14}{5}}$ .

$z^{\frac{4}{5}} (3 z^{\frac{10}{5}}) - 19 z^{\frac{9}{5}} + 28 z^{\frac{4}{5}} = 0$

Step 1.2

Factor $z^{\frac{4}{5}}$ out of $- 19 z^{\frac{9}{5}}$ .

$z^{\frac{4}{5}} (3 z^{\frac{10}{5}}) + z^{\frac{4}{5}} (- 19 z^{\frac{5}{5}}) + 28 z^{\frac{4}{5}} = 0$

Step 1.3

Factor $z^{\frac{4}{5}}$ out of $28 z^{\frac{4}{5}}$ .

$z^{\frac{4}{5}} (3 z^{\frac{10}{5}}) + z^{\frac{4}{5}} (- 19 z^{\frac{5}{5}}) + z^{\frac{4}{5}} \cdot 28 = 0$

Step 1.4

Factor $z^{\frac{4}{5}}$ out of $z^{\frac{4}{5}} (3 z^{\frac{10}{5}}) + z^{\frac{4}{5}} (- 19 z^{\frac{5}{5}})$ .

$z^{\frac{4}{5}} (3 z^{\frac{10}{5}} - 19 z^{\frac{5}{5}}) + z^{\frac{4}{5}} \cdot 28 = 0$

Step 1.5

Factor $z^{\frac{4}{5}}$ out of $z^{\frac{4}{5}} (3 z^{\frac{10}{5}} - 19 z^{\frac{5}{5}}) + z^{\frac{4}{5}} \cdot 28$ .

$z^{\frac{4}{5}} (3 z^{\frac{10}{5}} - 19 z^{\frac{5}{5}} + 28) = 0$

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 $a \cdot c = 3 \cdot 28 = 84$ and whose sum is $b = - 19$ .

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Step 2.1.1

Factor $- 19$ out of $- 19 z^{\frac{5}{5}}$ .

$z^{\frac{4}{5}} (3 z^{\frac{10}{5}} - 19 z^{\frac{5}{5}} + 28) = 0$

Step 2.1.2

Rewrite $- 19$ as $- 7$ plus $- 12$

$z^{\frac{4}{5}} (3 z^{\frac{10}{5}} + (- 7 - 12) z^{\frac{5}{5}} + 28) = 0$

Step 2.1.3

Apply the distributive property.

$z^{\frac{4}{5}} (3 z^{\frac{10}{5}} - 7 z^{\frac{5}{5}} - 12 z^{\frac{5}{5}} + 28) = 0$

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.

$z^{\frac{4}{5}} ((3 z^{\frac{10}{5}} - 7 z^{\frac{5}{5}}) - 12 z^{\frac{5}{5}} + 28) = 0$

Step 2.2.2

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

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

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 z^{\frac{5}{5}} - 7$ .

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

Step 3

Divide $5$ by $5$ .

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

Step 4

Divide $5$ by $5$ .

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

Step 5

Remove unnecessary parentheses.

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

Step 6

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

$z^{\frac{4}{5}} = 0$

$3 z - 7 = 0$

$z - 4 = 0$

Step 7

Set $z^{\frac{4}{5}}$ equal to $0$ and solve for $z$ .

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Step 7.1

Set $z^{\frac{4}{5}}$ equal to $0$ .

$z^{\frac{4}{5}} = 0$

Step 7.2

Solve $z^{\frac{4}{5}} = 0$ for $z$ .

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Step 7.2.1

Raise each side of the equation to the power of $\frac{5}{4}$ to eliminate the fractional exponent on the left side.

${(z^{\frac{4}{5}})}^{\frac{5}{4}} = \pm 0^{\frac{5}{4}}$

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 ${(z^{\frac{4}{5}})}^{\frac{5}{4}}$ .

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Step 7.2.2.1.1.1

Multiply the exponents in ${(z^{\frac{4}{5}})}^{\frac{5}{4}}$ .

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Step 7.2.2.1.1.1.1

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

$z^{\frac{4}{5} \cdot \frac{5}{4}} = \pm 0^{\frac{5}{4}}$

Step 7.2.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 7.2.2.1.1.1.2.1

Cancel the common factor.

$z^{\frac{4}{5} \cdot \frac{5}{4}} = \pm 0^{\frac{5}{4}}$

Step 7.2.2.1.1.1.2.2

Rewrite the expression.

$z^{\frac{1}{5} \cdot 5} = \pm 0^{\frac{5}{4}}$

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.

$z^{\frac{1}{5} \cdot 5} = \pm 0^{\frac{5}{4}}$

Step 7.2.2.1.1.1.3.2

Rewrite the expression.

$z^{1} = \pm 0^{\frac{5}{4}}$

Step 7.2.2.1.1.2

Simplify.

$z = \pm 0^{\frac{5}{4}}$

Step 7.2.2.2

Simplify the right side.

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Step 7.2.2.2.1

Simplify $\pm 0^{\frac{5}{4}}$ .

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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 $0^{4}$ .

$z = \pm {(0^{4})}^{\frac{5}{4}}$

Step 7.2.2.2.1.1.2

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

$z = \pm 0^{4 (\frac{5}{4})}$

Step 7.2.2.2.1.2

Cancel the common factor of $4$ .

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Step 7.2.2.2.1.2.1

Cancel the common factor.

$z = \pm 0^{4 (\frac{5}{4})}$

Step 7.2.2.2.1.2.2

Rewrite the expression.

$z = \pm 0^{5}$

Step 7.2.2.2.1.3

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

$z = \pm 0$

Step 7.2.2.2.1.4

Plus or minus $0$ is $0$ .

$z = 0$

Step 8

Set $3 z - 7$ equal to $0$ and solve for $z$ .

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Step 8.1

Set $3 z - 7$ equal to $0$ .

$3 z - 7 = 0$

Step 8.2

Solve $3 z - 7 = 0$ for $z$ .

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Step 8.2.1

Add $7$ to both sides of the equation.

$3 z = 7$

Step 8.2.2

Divide each term in $3 z = 7$ by $3$ and simplify.

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Step 8.2.2.1

Divide each term in $3 z = 7$ by $3$ .

$\frac{3 z}{3} = \frac{7}{3}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{7}{3}$

Step 8.2.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{7}{3}$

Step 9

Set $z - 4$ equal to $0$ and solve for $z$ .

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Step 9.1

Set $z - 4$ equal to $0$ .

$z - 4 = 0$

Step 9.2

Add $4$ to both sides of the equation.

$z = 4$

Step 10

The final solution is all the values that make $z^{\frac{4}{5}} (3 z - 7) (z - 4) = 0$ true.

$z = 0, \frac{7}{3}, 4$