Pre-Algebra Examples

$\sqrt{9 y^{7}}$

Step 1

Simplify $\sqrt{9 y^{7}}$ .

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Step 1.1

Rewrite $9 y^{7}$ as ${(3 y^{3})}^{2} y$ .

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Step 1.1.1

Rewrite $9$ as $3^{2}$ .

$x = \sqrt{3^{2} y^{7}}$

Step 1.1.2

Factor out $y^{6}$ .

$x = \sqrt{3^{2} (y^{6} y)}$

Step 1.1.3

Rewrite $y^{6}$ as ${(y^{3})}^{2}$ .

$x = \sqrt{3^{2} ({(y^{3})}^{2} y)}$

Step 1.1.4

Rewrite $3^{2} {(y^{3})}^{2}$ as ${(3 y^{3})}^{2}$ .

$x = \sqrt{{(3 y^{3})}^{2} y}$

Step 1.2

Pull terms out from under the radical.

$x = 3 y^{3} \sqrt{y}$

Step 2

The domain in terms of $y$ are all the $y$ values that make the radicand non-negative.

$[0, \infty)$

${y | y \geq 0}$

Step 3

To find the radical expression end point, substitute the $y$ value $0$ , which is the least value in the domain, into $f (y) = 3 y^{3} \sqrt{y}$ .

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Step 3.1

Replace the variable $y$ with $0$ in the expression.

$f (0) = 3 {(0)}^{3} \sqrt{0}$

Step 3.2

Simplify the result.

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Step 3.2.1

Remove parentheses.

$f (0) = 3 {(0)}^{3} \sqrt{0}$

Step 3.2.2

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

$f (0) = 3 \cdot (0 \sqrt{0})$

Step 3.2.3

Multiply $3$ by $0$ .

$f (0) = 0 \sqrt{0}$

Step 3.2.4

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

$f (0) = 0 \sqrt{0^{2}}$

Step 3.2.5

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

$f (0) = 0 \cdot 0$

Step 3.2.6

Multiply $0$ by $0$ .

$f (0) = 0$

Step 3.2.7

The final answer is $0$ .

$0$

Step 4

The radical expression end point is $(0, 0)$ .

$(0, 0)$

Step 5

Select a few $y$ values from the domain. It would be more useful to select the values so that they are next to the $y$ value of the radical expression end point.

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Step 5.1

Substitute the $y$ value $1$ into $3 y^{3} \sqrt{y}$ . In this case, the point is $(3, 1)$ .

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Step 5.1.1

Replace the variable $y$ with $1$ in the expression.

$f (1) = 3 {(1)}^{3} \sqrt{1}$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Remove parentheses.

$f (1) = 3 {(1)}^{3} \sqrt{1}$

Step 5.1.2.2

One to any power is one.

$f (1) = 3 \cdot (1 \sqrt{1})$

Step 5.1.2.3

Multiply $3$ by $1$ .

$f (1) = 3 \sqrt{1}$

Step 5.1.2.4

Any root of $1$ is $1$ .

$f (1) = 3 \cdot 1$

Step 5.1.2.5

Multiply $3$ by $1$ .

$f (1) = 3$

Step 5.1.2.6

The final answer is $3$ .

$y = 3$

Step 5.2

Substitute the $y$ value $2$ into $3 y^{3} \sqrt{y}$ . In this case, the point is $(33.94, 2)$ .

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Step 5.2.1

Replace the variable $y$ with $2$ in the expression.

$f (2) = 3 {(2)}^{3} \sqrt{2}$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Remove parentheses.

$f (2) = 3 {(2)}^{3} \sqrt{2}$

Step 5.2.2.2

Raise $2$ to the power of $3$ .

$f (2) = 3 \cdot (8 \sqrt{2})$

Step 5.2.2.3

Multiply $3$ by $8$ .

$f (2) = 24 \sqrt{2}$

Step 5.2.2.4

The final answer is $24 \sqrt{2}$ .

$y = 24 \sqrt{2}$

Step 5.3

The square root can be graphed using the points around the vertex $(0, 0), (3, 1), (33.94, 2)$

$\begin{array}{c} x & y \\ 0 & 0 \\ 3 & 1 \\ 33.94 & 2 \end{array}$

Step 6