Precalculus Examples

$n^{3} = - 75$ , $n^{6} = - 9375$

Step 1

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

$n = \sqrt[3]{- 75}$

$n^{6} = - 9375$

Step 2

Simplify $\sqrt[3]{- 75}$ .

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Step 2.1

Rewrite $- 75$ as ${(- 1)}^{3} \cdot 75$ .

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Step 2.1.1

Rewrite $- 75$ as $- 1 (75)$ .

$n = \sqrt[3]{- 1 \cdot 75}$

$n^{6} = - 9375$

Step 2.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$n = \sqrt[3]{{(- 1)}^{3} \cdot 75}$

$n^{6} = - 9375$

$n = \sqrt[3]{{(- 1)}^{3} \cdot 75}$

$n^{6} = - 9375$

Step 2.2

Pull terms out from under the radical.

$n = - 1 \sqrt[3]{75}$

$n^{6} = - 9375$

Step 2.3

Rewrite $- 1 \sqrt[3]{75}$ as $- \sqrt[3]{75}$ .

$n = - \sqrt[3]{75}$

$n^{6} = - 9375$

$n = - \sqrt[3]{75}$

$n^{6} = - 9375$

Step 3

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

$n = \pm \sqrt[6]{- 9375}$

$n = - \sqrt[3]{75}$

Step 4

Simplify $\pm \sqrt[6]{- 9375}$ .

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Step 4.1

Rewrite $- 9375$ as $i^{6} \cdot 9375$ .

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Step 4.1.1

Rewrite $- 9375$ as $- 1 (9375)$ .

$n = \pm \sqrt[6]{- 1 \cdot 9375}$

$n = - \sqrt[3]{75}$

Step 4.1.2

Rewrite $- 1$ as $i^{6}$ .

$n = \pm \sqrt[6]{i^{6} \cdot 9375}$

$n = - \sqrt[3]{75}$

$n = \pm \sqrt[6]{i^{6} \cdot 9375}$

$n = - \sqrt[3]{75}$

Step 4.2

Pull terms out from under the radical.

$n = \pm i \sqrt[6]{9375}$

$n = - \sqrt[3]{75}$

$n = \pm i \sqrt[6]{9375}$

$n = - \sqrt[3]{75}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$n = i \sqrt[6]{9375}$

$n = - \sqrt[3]{75}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$n = - i \sqrt[6]{9375}$

$n = - \sqrt[3]{75}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$n = i \sqrt[6]{9375}, - i \sqrt[6]{9375}$

$n = - \sqrt[3]{75}$

$n = i \sqrt[6]{9375}, - i \sqrt[6]{9375}$

$n = - \sqrt[3]{75}$

Step 6

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

$(n - (- \sqrt[3]{75})) (n - (i \sqrt[6]{9375}, - i \sqrt[6]{9375})) (n - (- i \sqrt[6]{9375}, - i \sqrt[6]{9375})) = 0$

Step 7

Simplify.

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Step 7.1

Multiply $n + \sqrt[3]{75}$ by each element of the matrix.

$((n + \sqrt[3]{75}) (n - i \sqrt[6]{9375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2

Simplify each element in the matrix.

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Step 7.2.1

Expand $(n + \sqrt[3]{75}) (n - i \sqrt[6]{9375})$ using the FOIL Method.

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Step 7.2.1.1

Apply the distributive property.

$(n (n - i \sqrt[6]{9375}) + \sqrt[3]{75} (n - i \sqrt[6]{9375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.1.2

Apply the distributive property.

$(n \cdot n + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} (n - i \sqrt[6]{9375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.1.3

Apply the distributive property.

$(n \cdot n + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n + \sqrt[3]{75} (- i \sqrt[6]{9375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2

Simplify each term.

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Step 7.2.2.1

Multiply $n$ by $n$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n + \sqrt[3]{75} (- i \sqrt[6]{9375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.2

Multiply $\sqrt[3]{75} (- i \sqrt[6]{9375})$ .

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Step 7.2.2.2.1

Rewrite the expression using the least common index of $6$ .

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Step 7.2.2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{75}$ as $75^{\frac{1}{3}}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i (75^{\frac{1}{3}} \sqrt[6]{9375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.2.1.2

Rewrite $75^{\frac{1}{3}}$ as $75^{\frac{2}{6}}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i (75^{\frac{2}{6}} \sqrt[6]{9375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.2.1.3

Rewrite $75^{\frac{2}{6}}$ as $\sqrt[6]{75^{2}}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i (\sqrt[6]{75^{2}} \sqrt[6]{9375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.2.2

Combine using the product rule for radicals.

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i \sqrt[6]{75^{2} \cdot 9375}, (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.2.3

Raise $75$ to the power of $2$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i \sqrt[6]{5625 \cdot 9375}, (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.2.4

Multiply $5625$ by $9375$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i \sqrt[6]{52734375}, (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.3

Rewrite $52734375$ as $5^{6} \cdot 3375$ .

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Step 7.2.2.3.1

Factor $15625$ out of $52734375$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i \sqrt[6]{15625 (3375)}, (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.3.2

Rewrite $15625$ as $5^{6}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i \sqrt[6]{5^{6} \cdot 3375}, (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.4

Pull terms out from under the radical.

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i (5 \sqrt[6]{3375}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.5

Rewrite $3375$ as $15^{3}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i (5 \sqrt[6]{15^{3}}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.6

Rewrite $\sqrt[6]{15^{3}}$ as $\sqrt{\sqrt[3]{15^{3}}}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i (5 \sqrt{\sqrt[3]{15^{3}}}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.7

Pull terms out from under the radical, assuming real numbers.

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - i (5 \sqrt{15}), (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.2.8

Multiply $5$ by $- 1$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, (n + \sqrt[3]{75}) (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.3

Apply the distributive property.

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) + \sqrt[3]{75} (- i \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.4

Multiply $\sqrt[3]{75} (- i \sqrt[6]{9375})$ .

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Step 7.2.4.1

Rewrite the expression using the least common index of $6$ .

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Step 7.2.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{75}$ as $75^{\frac{1}{3}}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i (75^{\frac{1}{3}} \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.4.1.2

Rewrite $75^{\frac{1}{3}}$ as $75^{\frac{2}{6}}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i (75^{\frac{2}{6}} \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.4.1.3

Rewrite $75^{\frac{2}{6}}$ as $\sqrt[6]{75^{2}}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i (\sqrt[6]{75^{2}} \sqrt[6]{9375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.4.2

Combine using the product rule for radicals.

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i \sqrt[6]{75^{2} \cdot 9375}) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.4.3

Raise $75$ to the power of $2$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i \sqrt[6]{5625 \cdot 9375}) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.4.4

Multiply $5625$ by $9375$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i \sqrt[6]{52734375}) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.5

Simplify each term.

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Step 7.2.5.1

Rewrite $52734375$ as $5^{6} \cdot 3375$ .

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Step 7.2.5.1.1

Factor $15625$ out of $52734375$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i \sqrt[6]{15625 (3375)}) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.5.1.2

Rewrite $15625$ as $5^{6}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i \sqrt[6]{5^{6} \cdot 3375}) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.5.2

Pull terms out from under the radical.

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i (5 \sqrt[6]{3375})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.5.3

Rewrite $3375$ as $15^{3}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i (5 \sqrt[6]{15^{3}})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.5.4

Rewrite $\sqrt[6]{15^{3}}$ as $\sqrt{\sqrt[3]{15^{3}}}$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i (5 \sqrt{\sqrt[3]{15^{3}}})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.5.5

Pull terms out from under the radical, assuming real numbers.

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - i (5 \sqrt{15})) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.5.6

Multiply $5$ by $- 1$ .

$(n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - 5 i \sqrt{15}) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$

Step 7.2.6

Reorder factors in $n^{2} + n (- i \sqrt[6]{9375}) + \sqrt[3]{75} n - 5 i \sqrt{15}, n (- i \sqrt[6]{9375}) - 5 i \sqrt{15}$ .

$(n^{2} - n i \sqrt[6]{9375} + \sqrt[3]{75} n - 5 i \sqrt{15}, - n i \sqrt[6]{9375} - 5 i \sqrt{15}) (n + i \sqrt[6]{9375}, - i \sqrt[6]{9375}) = 0$