Finite Math Examples

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[6]{4}}{\sqrt{x}}$

Step 1

Simplify

$\frac{\sqrt[6]{4}}{\sqrt{x}}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite

$4$ as

$2^{2}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[6]{2^{2}}}{\sqrt{x}}$

Step 1.1.2

Rewrite

$\sqrt[6]{2^{2}}$ as

$\sqrt[3]{\sqrt{2^{2}}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{\sqrt{2^{2}}}}{\sqrt{x}}$

Step 1.1.3

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

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2}}{\sqrt{x}}$

Step 1.2

Multiply

$\frac{\sqrt[3]{2}}{\sqrt{x}}$ by

$\frac{\sqrt{x}}{\sqrt{x}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply

$\frac{\sqrt[3]{2}}{\sqrt{x}}$ by

$\frac{\sqrt{x}}{\sqrt{x}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 1.3.2

Raise

$\sqrt{x}$ to the power of

$1$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 1.3.3

Raise

$\sqrt{x}$ to the power of

$1$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 1.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 1.3.5

Add

$1$ and

$1$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 1.3.6

Rewrite

${\sqrt{x}}^{2}$ as

$x$ .

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Step 1.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

$x^{\frac{1}{2}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 1.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 1.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 1.3.6.4

Cancel the common factor of

$2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 1.3.6.4.2

Rewrite the expression.

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{x^{1}}$

Step 1.3.6.5

Simplify.

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[3]{2} \sqrt{x}}{x}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Rewrite the expression using the least common index of

$6$ .

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Step 1.4.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{2}$ as

$2^{\frac{1}{3}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{2^{\frac{1}{3}} \sqrt{x}}{x}$

Step 1.4.1.2

Rewrite

$2^{\frac{1}{3}}$ as

$2^{\frac{2}{6}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{2^{\frac{2}{6}} \sqrt{x}}{x}$

Step 1.4.1.3

Rewrite

$2^{\frac{2}{6}}$ as

$\sqrt[6]{2^{2}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[6]{2^{2}} \sqrt{x}}{x}$

Step 1.4.1.4

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

$x^{\frac{1}{2}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[6]{2^{2}} x^{\frac{1}{2}}}{x}$

Step 1.4.1.5

Rewrite

$x^{\frac{1}{2}}$ as

$x^{\frac{3}{6}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[6]{2^{2}} x^{\frac{3}{6}}}{x}$

Step 1.4.1.6

Rewrite

$x^{\frac{3}{6}}$ as

$\sqrt[6]{x^{3}}$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[6]{2^{2}} \sqrt[6]{x^{3}}}{x}$

Step 1.4.2

Combine using the product rule for radicals.

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[6]{2^{2} x^{3}}}{x}$

Step 1.4.3

Raise

$2$ to the power of

$2$ .

$\sqrt[4]{- 6^{x - 2}} = \frac{\sqrt[6]{4 x^{3}}}{x}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution