Trigonometry Examples

${(9 x^{2} y^{6})}^{- \frac{1}{2}}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(9 x^{2} y^{6})}^{\frac{1}{2}}}$

Step 1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{\sqrt{{(9 x^{2} y^{6})}^{1}}}$

Step 1.3

Anything raised to $1$ is the base itself.

$\frac{1}{\sqrt{9 x^{2} y^{6}}}$

Step 2

Set the radicand in $\sqrt{9 x^{2} y^{6}}$ greater than or equal to $0$ to find where the expression is defined.

$9 x^{2} y^{6} \geq 0$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $9 x^{2} y^{6} \geq 0$ by $9 y^{6}$ and simplify.

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Step 3.1.1

Divide each term in $9 x^{2} y^{6} \geq 0$ by $9 y^{6}$ .

$\frac{9 x^{2} y^{6}}{9 y^{6}} \geq \frac{0}{9 y^{6}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $9$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{9 x^{2} y^{6}}{9 y^{6}} \geq \frac{0}{9 y^{6}}$

Step 3.1.2.1.2

Rewrite the expression.

$\frac{x^{2} y^{6}}{y^{6}} \geq \frac{0}{9 y^{6}}$

Step 3.1.2.2

Cancel the common factor of $y^{6}$ .

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Step 3.1.2.2.1

Cancel the common factor.

$\frac{x^{2} y^{6}}{y^{6}} \geq \frac{0}{9 y^{6}}$

Step 3.1.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \geq \frac{0}{9 y^{6}}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Cancel the common factor of $0$ and $9$ .

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Step 3.1.3.1.1

Factor $9$ out of $0$ .

$x^{2} \geq \frac{9 (0)}{9 y^{6}}$

Step 3.1.3.1.2

Cancel the common factors.

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Step 3.1.3.1.2.1

Factor $9$ out of $9 y^{6}$ .

$x^{2} \geq \frac{9 (0)}{9 (y^{6})}$

Step 3.1.3.1.2.2

Cancel the common factor.

$x^{2} \geq \frac{9 \cdot 0}{9 y^{6}}$

Step 3.1.3.1.2.3

Rewrite the expression.

$x^{2} \geq \frac{0}{y^{6}}$

Step 3.1.3.2

Divide $0$ by $y^{6}$ .

$x^{2} \geq 0$

Step 3.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 4

Set the denominator in $\frac{1}{\sqrt{9 x^{2} y^{6}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{9 x^{2} y^{6}} = 0$

Step 5

Solve for $x$ .

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Step 5.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{9 x^{2} y^{6}}}^{2} = 0^{2}$

Step 5.2

Simplify each side of the equation.

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Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x^{2} y^{6}}$ as ${(9 x^{2} y^{6})}^{\frac{1}{2}}$ .

${({(9 x^{2} y^{6})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Simplify ${({(9 x^{2} y^{6})}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.2.1.1

Multiply the exponents in ${({(9 x^{2} y^{6})}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.2.1.1.1

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

${(9 x^{2} y^{6})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 5.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.2.1.1.2.1

Cancel the common factor.

${(9 x^{2} y^{6})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 5.2.2.1.1.2.2

Rewrite the expression.

${(9 x^{2} y^{6})}^{1} = 0^{2}$

Step 5.2.2.1.2

Simplify.

$9 x^{2} y^{6} = 0^{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

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

$9 x^{2} y^{6} = 0$

Step 5.3

Solve for $x$ .

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Step 5.3.1

Divide each term in $9 x^{2} y^{6} = 0$ by $9 y^{6}$ and simplify.

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Step 5.3.1.1

Divide each term in $9 x^{2} y^{6} = 0$ by $9 y^{6}$ .

$\frac{9 x^{2} y^{6}}{9 y^{6}} = \frac{0}{9 y^{6}}$

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor of $9$ .

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Step 5.3.1.2.1.1

Cancel the common factor.

$\frac{9 x^{2} y^{6}}{9 y^{6}} = \frac{0}{9 y^{6}}$

Step 5.3.1.2.1.2

Rewrite the expression.

$\frac{x^{2} y^{6}}{y^{6}} = \frac{0}{9 y^{6}}$

Step 5.3.1.2.2

Cancel the common factor of $y^{6}$ .

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Step 5.3.1.2.2.1

Cancel the common factor.

$\frac{x^{2} y^{6}}{y^{6}} = \frac{0}{9 y^{6}}$

Step 5.3.1.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{9 y^{6}}$

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Cancel the common factor of $0$ and $9$ .

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Step 5.3.1.3.1.1

Factor $9$ out of $0$ .

$x^{2} = \frac{9 (0)}{9 y^{6}}$

Step 5.3.1.3.1.2

Cancel the common factors.

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Step 5.3.1.3.1.2.1

Factor $9$ out of $9 y^{6}$ .

$x^{2} = \frac{9 (0)}{9 (y^{6})}$

Step 5.3.1.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{9 \cdot 0}{9 y^{6}}$

Step 5.3.1.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{0}{y^{6}}$

Step 5.3.1.3.2

Divide $0$ by $y^{6}$ .

$x^{2} = 0$

Step 5.3.2

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

$x = \pm \sqrt{0}$

Step 5.3.3

Simplify $\pm \sqrt{0}$ .

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Step 5.3.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 5.3.3.2

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

$x = \pm 0$

Step 5.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$