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