Finite Math Examples

$2 x^{\frac{11}{5}} - 19 x^{\frac{6}{5}} + 24 x^{\frac{1}{5}} = 0$

Step 1

Find a common factor $x^{\frac{1}{5}}$ that is present in each term.

$2 {(x^{\frac{1}{5}})}^{11} - 19 {(x^{\frac{1}{5}})}^{6} + 24 x^{\frac{1}{5}}$

Step 2

Substitute $u$ for $x^{\frac{1}{5}}$ .

$2 {(u)}^{11} - 19 {(u)}^{6} + 24 (u) = 0$

Step 3

Solve for $u$ .

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Step 3.1

Multiply $24$ by $u$ .

$2 u^{11} - 19 u^{6} + 24 u = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $u$ out of $2 u^{11} - 19 u^{6} + 24 u$ .

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Step 3.2.1.1

Factor $u$ out of $2 u^{11}$ .

$u (2 u^{10}) - 19 u^{6} + 24 u = 0$

Step 3.2.1.2

Factor $u$ out of $- 19 u^{6}$ .

$u (2 u^{10}) + u (- 19 u^{5}) + 24 u = 0$

Step 3.2.1.3

Factor $u$ out of $24 u$ .

$u (2 u^{10}) + u (- 19 u^{5}) + u \cdot 24 = 0$

Step 3.2.1.4

Factor $u$ out of $u (2 u^{10}) + u (- 19 u^{5})$ .

$u (2 u^{10} - 19 u^{5}) + u \cdot 24 = 0$

Step 3.2.1.5

Factor $u$ out of $u (2 u^{10} - 19 u^{5}) + u \cdot 24$ .

$u (2 u^{10} - 19 u^{5} + 24) = 0$

Step 3.2.2

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

$u (2 {(u^{5})}^{2} - 19 u^{5} + 24) = 0$

Step 3.2.3

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

$u (2 u^{2} - 19 u + 24) = 0$

Step 3.2.4

Factor by grouping.

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Step 3.2.4.1

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 $a \cdot c = 2 \cdot 24 = 48$ and whose sum is $b = - 19$ .

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Step 3.2.4.1.1

Factor $- 19$ out of $- 19 u$ .

$u (2 u^{2} - 19 u + 24) = 0$

Step 3.2.4.1.2

Rewrite $- 19$ as $- 3$ plus $- 16$

$u (2 u^{2} + (- 3 - 16) u + 24) = 0$

Step 3.2.4.1.3

Apply the distributive property.

$u (2 u^{2} - 3 u - 16 u + 24) = 0$

Step 3.2.4.2

Factor out the greatest common factor from each group.

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Step 3.2.4.2.1

Group the first two terms and the last two terms.

$u ((2 u^{2} - 3 u) - 16 u + 24) = 0$

Step 3.2.4.2.2

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

$u (u (2 u - 3) - 8 (2 u - 3)) = 0$

Step 3.2.4.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 3$ .

$u ((2 u - 3) (u - 8)) = 0$

Step 3.2.5

Factor.

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Step 3.2.5.1

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

$u ((2 u^{5} - 3) (u^{5} - 8)) = 0$

Step 3.2.5.2

Remove unnecessary parentheses.

$u (2 u^{5} - 3) (u^{5} - 8) = 0$

Step 3.3

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

$u = 0$

$2 u^{5} - 3 = 0$

$u^{5} - 8 = 0$

Step 3.4

Set $u$ equal to $0$ .

$u = 0$

Step 3.5

Set $2 u^{5} - 3$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $2 u^{5} - 3$ equal to $0$ .

$2 u^{5} - 3 = 0$

Step 3.5.2

Solve $2 u^{5} - 3 = 0$ for $u$ .

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Step 3.5.2.1

Add $3$ to both sides of the equation.

$2 u^{5} = 3$

Step 3.5.2.2

Divide each term in $2 u^{5} = 3$ by $2$ and simplify.

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Step 3.5.2.2.1

Divide each term in $2 u^{5} = 3$ by $2$ .

$\frac{2 u^{5}}{2} = \frac{3}{2}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 u^{5}}{2} = \frac{3}{2}$

Step 3.5.2.2.2.1.2

Divide $u^{5}$ by $1$ .

$u^{5} = \frac{3}{2}$

Step 3.5.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \sqrt[5]{\frac{3}{2}}$

Step 3.5.2.4

Simplify $\sqrt[5]{\frac{3}{2}}$ .

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Step 3.5.2.4.1

Rewrite $\sqrt[5]{\frac{3}{2}}$ as $\frac{\sqrt[5]{3}}{\sqrt[5]{2}}$ .

$u = \frac{\sqrt[5]{3}}{\sqrt[5]{2}}$

Step 3.5.2.4.2

Multiply $\frac{\sqrt[5]{3}}{\sqrt[5]{2}}$ by $\frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$ .

$u = \frac{\sqrt[5]{3}}{\sqrt[5]{2}} \cdot \frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$

Step 3.5.2.4.3

Combine and simplify the denominator.

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Step 3.5.2.4.3.1

Multiply $\frac{\sqrt[5]{3}}{\sqrt[5]{2}}$ by $\frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$ .

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{\sqrt[5]{2} {\sqrt[5]{2}}^{4}}$

Step 3.5.2.4.3.2

Raise $\sqrt[5]{2}$ to the power of $1$ .

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{1} {\sqrt[5]{2}}^{4}}$

Step 3.5.2.4.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{1 + 4}}$

Step 3.5.2.4.3.4

Add $1$ and $4$ .

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{5}}$

Step 3.5.2.4.3.5

Rewrite ${\sqrt[5]{2}}^{5}$ as $2$ .

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Step 3.5.2.4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2}$ as $2^{\frac{1}{5}}$ .

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{{(2^{\frac{1}{5}})}^{5}}$

Step 3.5.2.4.3.5.2

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

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{2^{\frac{1}{5} \cdot 5}}$

Step 3.5.2.4.3.5.3

Combine $\frac{1}{5}$ and $5$ .

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{2^{\frac{5}{5}}}$

Step 3.5.2.4.3.5.4

Cancel the common factor of $5$ .

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Step 3.5.2.4.3.5.4.1

Cancel the common factor.

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{2^{\frac{5}{5}}}$

Step 3.5.2.4.3.5.4.2

Rewrite the expression.

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{2^{1}}$

Step 3.5.2.4.3.5.5

Evaluate the exponent.

$u = \frac{\sqrt[5]{3} {\sqrt[5]{2}}^{4}}{2}$

Step 3.5.2.4.4

Simplify the numerator.

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Step 3.5.2.4.4.1

Rewrite ${\sqrt[5]{2}}^{4}$ as $\sqrt[5]{2^{4}}$ .

$u = \frac{\sqrt[5]{3} \sqrt[5]{2^{4}}}{2}$

Step 3.5.2.4.4.2

Raise $2$ to the power of $4$ .

$u = \frac{\sqrt[5]{3} \sqrt[5]{16}}{2}$

Step 3.5.2.4.5

Simplify the numerator.

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Step 3.5.2.4.5.1

Combine using the product rule for radicals.

$u = \frac{\sqrt[5]{3 \cdot 16}}{2}$

Step 3.5.2.4.5.2

Multiply $3$ by $16$ .

$u = \frac{\sqrt[5]{48}}{2}$

Step 3.6

Set $u^{5} - 8$ equal to $0$ and solve for $u$ .

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Step 3.6.1

Set $u^{5} - 8$ equal to $0$ .

$u^{5} - 8 = 0$

Step 3.6.2

Solve $u^{5} - 8 = 0$ for $u$ .

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Step 3.6.2.1

Add $8$ to both sides of the equation.

$u^{5} = 8$

Step 3.6.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \sqrt[5]{8}$

Step 3.7

The final solution is all the values that make $u (2 u^{5} - 3) (u^{5} - 8) = 0$ true.

$u = 0, \frac{\sqrt[5]{48}}{2}, \sqrt[5]{8}$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{1}{5}} = 0, \frac{\sqrt[5]{48}}{2}, \sqrt[5]{8}$

Step 5

Solve for $x^{\frac{1}{5}} = 0$ for $x$ .

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Step 5.1

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{5}})}^{5} = 0^{5}$

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 ${(x^{\frac{1}{5}})}^{5}$ .

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Step 5.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 5.2.1.1.1.1

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

$x^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 5.2.1.1.1.2

Cancel the common factor of $5$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{5}$

Step 5.2.1.1.2

Simplify.

$x = 0^{5}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

$x = 0$

Step 6

Solve for $x^{\frac{1}{5}} = \frac{\sqrt[5]{48}}{2}$ for $x$ .

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Step 6.1

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{5}})}^{5} = {(\frac{\sqrt[5]{48}}{2})}^{5}$

Step 6.2

Simplify the exponent.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify ${(x^{\frac{1}{5}})}^{5}$ .

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Step 6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 6.2.1.1.1.1

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

$x^{\frac{1}{5} \cdot 5} = {(\frac{\sqrt[5]{48}}{2})}^{5}$

Step 6.2.1.1.1.2

Cancel the common factor of $5$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = {(\frac{\sqrt[5]{48}}{2})}^{5}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{\sqrt[5]{48}}{2})}^{5}$

Step 6.2.1.1.2

Simplify.

$x = {(\frac{\sqrt[5]{48}}{2})}^{5}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify ${(\frac{\sqrt[5]{48}}{2})}^{5}$ .

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Step 6.2.2.1.1

Apply the product rule to $\frac{\sqrt[5]{48}}{2}$ .

$x = \frac{{\sqrt[5]{48}}^{5}}{2^{5}}$

Step 6.2.2.1.2

Rewrite ${\sqrt[5]{48}}^{5}$ as $48$ .

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Step 6.2.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{48}$ as $48^{\frac{1}{5}}$ .

$x = \frac{{(48^{\frac{1}{5}})}^{5}}{2^{5}}$

Step 6.2.2.1.2.2

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

$x = \frac{48^{\frac{1}{5} \cdot 5}}{2^{5}}$

Step 6.2.2.1.2.3

Combine $\frac{1}{5}$ and $5$ .

$x = \frac{48^{\frac{5}{5}}}{2^{5}}$

Step 6.2.2.1.2.4

Cancel the common factor of $5$ .

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Step 6.2.2.1.2.4.1

Cancel the common factor.

$x = \frac{48^{\frac{5}{5}}}{2^{5}}$

Step 6.2.2.1.2.4.2

Rewrite the expression.

$x = \frac{48^{1}}{2^{5}}$

Step 6.2.2.1.2.5

Evaluate the exponent.

$x = \frac{48}{2^{5}}$

Step 6.2.2.1.3

Raise $2$ to the power of $5$ .

$x = \frac{48}{32}$

Step 6.2.2.1.4

Cancel the common factor of $48$ and $32$ .

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Step 6.2.2.1.4.1

Factor $16$ out of $48$ .

$x = \frac{16 (3)}{32}$

Step 6.2.2.1.4.2

Cancel the common factors.

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Step 6.2.2.1.4.2.1

Factor $16$ out of $32$ .

$x = \frac{16 \cdot 3}{16 \cdot 2}$

Step 6.2.2.1.4.2.2

Cancel the common factor.

$x = \frac{16 \cdot 3}{16 \cdot 2}$

Step 6.2.2.1.4.2.3

Rewrite the expression.

$x = \frac{3}{2}$

Step 7

Solve for $x^{\frac{1}{5}} = \sqrt[5]{8}$ for $x$ .

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Step 7.1

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{5}})}^{5} = {\sqrt[5]{8}}^{5}$

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 ${(x^{\frac{1}{5}})}^{5}$ .

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Step 7.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 7.2.1.1.1.1

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

$x^{\frac{1}{5} \cdot 5} = {\sqrt[5]{8}}^{5}$

Step 7.2.1.1.1.2

Cancel the common factor of $5$ .

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Step 7.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = {\sqrt[5]{8}}^{5}$

Step 7.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {\sqrt[5]{8}}^{5}$

Step 7.2.1.1.2

Simplify.

$x = {\sqrt[5]{8}}^{5}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Rewrite ${\sqrt[5]{8}}^{5}$ as $8$ .

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Step 7.2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8}$ as $8^{\frac{1}{5}}$ .

$x = {(8^{\frac{1}{5}})}^{5}$

Step 7.2.2.1.2

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

$x = 8^{\frac{1}{5} \cdot 5}$

Step 7.2.2.1.3

Combine $\frac{1}{5}$ and $5$ .

$x = 8^{\frac{5}{5}}$

Step 7.2.2.1.4

Cancel the common factor of $5$ .

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Step 7.2.2.1.4.1

Cancel the common factor.

$x = 8^{\frac{5}{5}}$

Step 7.2.2.1.4.2

Rewrite the expression.

$x = 8^{1}$

Step 7.2.2.1.5

Evaluate the exponent.

$x = 8$

Step 8

List all of the solutions.

$x = 0, \frac{3}{2}, 8$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{3}{2}, 8$

Decimal Form:

$x = 0, 1.5, 8$

Mixed Number Form:

$x = 0, 1 \frac{1}{2}, 8$