Finite Math Examples

$2 z^{\frac{13}{5}} - 14 z^{\frac{8}{5}} + 12 z^{\frac{3}{5}} = 0$

Step 1

Find a common factor $z^{\frac{3}{5}}$ that is present in each term.

$2 {(z^{\frac{3}{5}})}^{\frac{13}{3}} - 14 {(z^{\frac{3}{5}})}^{\frac{8}{3}} + 12 z^{\frac{3}{5}}$

Step 2

Substitute $u$ for $z^{\frac{3}{5}}$ .

$2 {(u)}^{\frac{13}{3}} - 14 {(u)}^{\frac{8}{3}} + 12 (u) = 0$

Step 3

Solve for $u$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Factor $2 u$ out of $2 u^{\frac{13}{3}} - 14 u^{\frac{8}{3}} + 12 u$ .

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Step 3.1.1.1

Factor $2 u$ out of $2 u^{\frac{13}{3}}$ .

$2 u (u^{\frac{10}{3}}) - 14 u^{\frac{8}{3}} + 12 u = 0$

Step 3.1.1.2

Factor $2 u$ out of $- 14 u^{\frac{8}{3}}$ .

$2 u (u^{\frac{10}{3}}) + 2 u (- 7 u^{\frac{5}{3}}) + 12 u = 0$

Step 3.1.1.3

Factor $2 u$ out of $12 u$ .

$2 u (u^{\frac{10}{3}}) + 2 u (- 7 u^{\frac{5}{3}}) + 2 u (6) = 0$

Step 3.1.1.4

Factor $2 u$ out of $2 u (u^{\frac{10}{3}}) + 2 u (- 7 u^{\frac{5}{3}})$ .

$2 u (u^{\frac{10}{3}} - 7 u^{\frac{5}{3}}) + 2 u (6) = 0$

Step 3.1.1.5

Factor $2 u$ out of $2 u (u^{\frac{10}{3}} - 7 u^{\frac{5}{3}}) + 2 u (6)$ .

$2 u (u^{\frac{10}{3}} - 7 u^{\frac{5}{3}} + 6) = 0$

Step 3.1.2

Rewrite $u^{\frac{10}{3}}$ as ${(u^{\frac{5}{3}})}^{2}$ .

$2 u ({(u^{\frac{5}{3}})}^{2} - 7 u^{\frac{5}{3}} + 6) = 0$

Step 3.1.3

Let $u = u^{\frac{5}{3}}$ . Substitute $u$ for all occurrences of $u^{\frac{5}{3}}$ .

$2 u (u^{2} - 7 u + 6) = 0$

Step 3.1.4

Factor $u^{2} - 7 u + 6$ using the AC method.

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Step 3.1.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 3.1.4.2

Write the factored form using these integers.

$2 u ((u - 6) (u - 1)) = 0$

Step 3.1.5

Factor.

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Step 3.1.5.1

Replace all occurrences of $u$ with $u^{\frac{5}{3}}$ .

$2 u ((u^{\frac{5}{3}} - 6) (u^{\frac{5}{3}} - 1)) = 0$

Step 3.1.5.2

Remove unnecessary parentheses.

$2 u (u^{\frac{5}{3}} - 6) (u^{\frac{5}{3}} - 1) = 0$

Step 3.2

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

$u = 0$

$u^{\frac{5}{3}} - 6 = 0$

$u^{\frac{5}{3}} - 1 = 0$

Step 3.3

Set $u$ equal to $0$ .

$u = 0$

Step 3.4

Set $u^{\frac{5}{3}} - 6$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $u^{\frac{5}{3}} - 6$ equal to $0$ .

$u^{\frac{5}{3}} - 6 = 0$

Step 3.4.2

Solve $u^{\frac{5}{3}} - 6 = 0$ for $u$ .

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Step 3.4.2.1

Add $6$ to both sides of the equation.

$u^{\frac{5}{3}} = 6$

Step 3.4.2.2

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

${(u^{\frac{5}{3}})}^{\frac{3}{5}} = 6^{\frac{3}{5}}$

Step 3.4.2.3

Simplify the left side.

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Step 3.4.2.3.1

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

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Step 3.4.2.3.1.1

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

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Step 3.4.2.3.1.1.1

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

$u^{\frac{5}{3} \cdot \frac{3}{5}} = 6^{\frac{3}{5}}$

Step 3.4.2.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.4.2.3.1.1.2.1

Cancel the common factor.

$u^{\frac{5}{3} \cdot \frac{3}{5}} = 6^{\frac{3}{5}}$

Step 3.4.2.3.1.1.2.2

Rewrite the expression.

$u^{\frac{1}{3} \cdot 3} = 6^{\frac{3}{5}}$

Step 3.4.2.3.1.1.3

Cancel the common factor of $3$ .

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Step 3.4.2.3.1.1.3.1

Cancel the common factor.

$u^{\frac{1}{3} \cdot 3} = 6^{\frac{3}{5}}$

Step 3.4.2.3.1.1.3.2

Rewrite the expression.

$u^{1} = 6^{\frac{3}{5}}$

Step 3.4.2.3.1.2

Simplify.

$u = 6^{\frac{3}{5}}$

Step 3.5

Set $u^{\frac{5}{3}} - 1$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $u^{\frac{5}{3}} - 1$ equal to $0$ .

$u^{\frac{5}{3}} - 1 = 0$

Step 3.5.2

Solve $u^{\frac{5}{3}} - 1 = 0$ for $u$ .

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Step 3.5.2.1

Add $1$ to both sides of the equation.

$u^{\frac{5}{3}} = 1$

Step 3.5.2.2

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

${(u^{\frac{5}{3}})}^{\frac{3}{5}} = 1^{\frac{3}{5}}$

Step 3.5.2.3

Simplify the exponent.

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Step 3.5.2.3.1

Simplify the left side.

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Step 3.5.2.3.1.1

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

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Step 3.5.2.3.1.1.1

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

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Step 3.5.2.3.1.1.1.1

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

$u^{\frac{5}{3} \cdot \frac{3}{5}} = 1^{\frac{3}{5}}$

Step 3.5.2.3.1.1.1.2

Cancel the common factor of $5$ .

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Step 3.5.2.3.1.1.1.2.1

Cancel the common factor.

$u^{\frac{5}{3} \cdot \frac{3}{5}} = 1^{\frac{3}{5}}$

Step 3.5.2.3.1.1.1.2.2

Rewrite the expression.

$u^{\frac{1}{3} \cdot 3} = 1^{\frac{3}{5}}$

Step 3.5.2.3.1.1.1.3

Cancel the common factor of $3$ .

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Step 3.5.2.3.1.1.1.3.1

Cancel the common factor.

$u^{\frac{1}{3} \cdot 3} = 1^{\frac{3}{5}}$

Step 3.5.2.3.1.1.1.3.2

Rewrite the expression.

$u^{1} = 1^{\frac{3}{5}}$

Step 3.5.2.3.1.1.2

Simplify.

$u = 1^{\frac{3}{5}}$

Step 3.5.2.3.2

Simplify the right side.

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Step 3.5.2.3.2.1

One to any power is one.

$u = 1$

Step 3.6

The final solution is all the values that make $2 u (u^{\frac{5}{3}} - 6) (u^{\frac{5}{3}} - 1) = 0$ true.

$u = 0, 6^{\frac{3}{5}}, 1$

Step 4

Substitute $z$ for $u$ .

$z^{\frac{3}{5}} = 0, 6^{\frac{3}{5}}, 1$

Step 5

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

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Step 5.1

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 5.2

Simplify the exponent.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

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 5.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.1.2.2

Rewrite the expression.

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

Step 5.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 5.2.1.1.1.3.1

Cancel the common factor.

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

Step 5.2.1.1.1.3.2

Rewrite the expression.

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

Step 5.2.1.1.2

Simplify.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Simplify the expression.

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Step 5.2.2.1.1.1

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

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

Step 5.2.2.1.1.2

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

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

Step 5.2.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.2.1.2.1

Cancel the common factor.

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

Step 5.2.2.1.2.2

Rewrite the expression.

$z = 0^{5}$

Step 5.2.2.1.3

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

$z = 0$

Step 6

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$z = 6$

Step 7

Solve for $z^{\frac{3}{5}} = 1$ for $z$ .

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Step 7.1

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

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

Step 7.2

Simplify the exponent.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

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

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Step 7.2.1.1.1

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

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Step 7.2.1.1.1.1

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

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

Step 7.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 7.2.1.1.1.2.1

Cancel the common factor.

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

Step 7.2.1.1.1.2.2

Rewrite the expression.

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

Step 7.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 7.2.1.1.1.3.1

Cancel the common factor.

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

Step 7.2.1.1.1.3.2

Rewrite the expression.

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

Step 7.2.1.1.2

Simplify.

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

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

One to any power is one.

$z = 1$

Step 8

List all of the solutions.

$z = 0, 6, 1$