Pre-Algebra Examples

$\sqrt{x^{6} y^{7}}$

Step 1

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

${\sqrt{x^{6} y^{7}}}^{2} = 0^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{6} y^{7}}$ as ${(x^{6} y^{7})}^{\frac{1}{2}}$ .

${({(x^{6} y^{7})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{6} y^{7})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{6} y^{7})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x^{6} y^{7})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x^{6} y^{7})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x^{6} y^{7})}^{1} = 0^{2}$

Step 2.2.1.2

Simplify.

$x^{6} y^{7} = 0^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$x^{6} y^{7} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $x^{6} y^{7} = 0$ by $y^{7}$ and simplify.

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Step 3.1.1

Divide each term in $x^{6} y^{7} = 0$ by $y^{7}$ .

$\frac{x^{6} y^{7}}{y^{7}} = \frac{0}{y^{7}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $y^{7}$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{x^{6} y^{7}}{y^{7}} = \frac{0}{y^{7}}$

Step 3.1.2.1.2

Divide $x^{6}$ by $1$ .

$x^{6} = \frac{0}{y^{7}}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $y^{7}$ .

$x^{6} = 0$

Step 3.2

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

$x = \pm \sqrt[6]{0}$

Step 3.3

Simplify $\pm \sqrt[6]{0}$ .

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Step 3.3.1

Rewrite $0$ as $0^{6}$ .

$x = \pm \sqrt[6]{0^{6}}$

Step 3.3.2

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

$x = \pm 0$

Step 3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

To find the end points, substitute the bounds of the $y$ values from the domain into $f (y) = 0$ .

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Step 4.1

Replace the variable $y$ with $S E T, x | x E R$ in the expression.

$f (S E T, x | x E R) = 0$

Step 4.2

The final answer is $0$ .

$0$

Step 4.3

Replace the variable $y$ with $\infty$ in the expression.

$f (\infty) = 0$

Step 4.4

The final answer is $0$ .

$0$

Step 5

The end points are $(0, S E T, x | x E R), (0, \infty)$ .

$(0, S E T, x | x E R), (0, \infty)$

Step 6

The square root can be graphed using the points around the vertex $(0, S E T, x | x E R), (0, \infty),$

$\begin{array}{c} x & y \\ 0 & S E \\ 0 & \infty \end{array}$

Step 7