Pre-Algebra Examples

f (x) = \sqrt{x^{4} - 81}

$f (x) = \sqrt{x^{4} - 81}$

Step 1

Find the

y

$y$ value at

x = - 3

$x = - 3$ .

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Step 1.1

Replace the variable

x

$x$ with

- 3

$- 3$ in the expression.

f (- 3) = \sqrt{({(- 3)}^{2} + 9) ((- 3) + 3) ((- 3) - 3)}

$f (- 3) = \sqrt{({(- 3)}^{2} + 9) ((- 3) + 3) ((- 3) - 3)}$

Step 1.2

Simplify the result.

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Step 1.2.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

f (- 3) = \sqrt{(9 + 9) (- 3 + 3) (- 3 - 3)}

$f (- 3) = \sqrt{(9 + 9) (- 3 + 3) (- 3 - 3)}$

Step 1.2.2

Add

9

$9$ and

9

$9$ .

f (- 3) = \sqrt{18 (- 3 + 3) (- 3 - 3)}

$f (- 3) = \sqrt{18 (- 3 + 3) (- 3 - 3)}$

Step 1.2.3

Add

- 3

$- 3$ and

3

$3$ .

f (- 3) = \sqrt{18 \cdot (0 (- 3 - 3))}

$f (- 3) = \sqrt{18 \cdot (0 (- 3 - 3))}$

Step 1.2.4

Combine exponents.

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Step 1.2.4.1

Multiply

18

$18$ by

0

$0$ .

f (- 3) = \sqrt{0 (- 3 - 3)}

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

Step 1.2.4.2

Multiply

0

$0$ by

- 3 - 3

$- 3 - 3$ .

f (- 3) = \sqrt{0}

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

f (- 3) = \sqrt{0}

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

Step 1.2.5

Rewrite

0

$0$ as

0^{2}

$0^{2}$ .

f (- 3) = \sqrt{0^{2}}

$f (- 3) = \sqrt{0^{2}}$

Step 1.2.6

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

f (- 3) = 0

$f (- 3) = 0$

Step 1.2.7

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 1.3

The

y

$y$ value at

x = - 3

$x = - 3$ is

0

$0$ .

y = 0

$y = 0$

y = 0

$y = 0$

Step 2

Find the

y

$y$ value at

x = - 4

$x = - 4$ .

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Step 2.1

Replace the variable

x

$x$ with

- 4

$- 4$ in the expression.

f (- 4) = \sqrt{({(- 4)}^{2} + 9) ((- 4) + 3) ((- 4) - 3)}

$f (- 4) = \sqrt{({(- 4)}^{2} + 9) ((- 4) + 3) ((- 4) - 3)}$

Step 2.2

Simplify the result.

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Step 2.2.1

Raise

- 4

$- 4$ to the power of

2

$2$ .

f (- 4) = \sqrt{(16 + 9) (- 4 + 3) (- 4 - 3)}

$f (- 4) = \sqrt{(16 + 9) (- 4 + 3) (- 4 - 3)}$

Step 2.2.2

Add

16

$16$ and

9

$9$ .

f (- 4) = \sqrt{25 (- 4 + 3) (- 4 - 3)}

$f (- 4) = \sqrt{25 (- 4 + 3) (- 4 - 3)}$

Step 2.2.3

Add

- 4

$- 4$ and

3

$3$ .

f (- 4) = \sqrt{25 \cdot (- 1 (- 4 - 3))}

$f (- 4) = \sqrt{25 \cdot (- 1 (- 4 - 3))}$

Step 2.2.4

Combine exponents.

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Step 2.2.4.1

Factor out negative.

f (- 4) = \sqrt{- (25 (- 4 - 3))}

$f (- 4) = \sqrt{- (25 (- 4 - 3))}$

Step 2.2.4.2

Multiply

25

$25$ by

- 1

$- 1$ .

f (- 4) = \sqrt{- 25 (- 4 - 3)}

$f (- 4) = \sqrt{- 25 (- 4 - 3)}$

f (- 4) = \sqrt{- 25 (- 4 - 3)}

$f (- 4) = \sqrt{- 25 (- 4 - 3)}$

Step 2.2.5

Subtract

3

$3$ from

- 4

$- 4$ .

f (- 4) = \sqrt{- 25 \cdot - 7}

$f (- 4) = \sqrt{- 25 \cdot - 7}$

Step 2.2.6

Multiply

- 25

$- 25$ by

- 7

$- 7$ .

f (- 4) = \sqrt{175}

$f (- 4) = \sqrt{175}$

Step 2.2.7

Rewrite

175

$175$ as

5^{2} \cdot 7

$5^{2} \cdot 7$ .

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Step 2.2.7.1

Factor

25

$25$ out of

175

$175$ .

f (- 4) = \sqrt{25 (7)}

$f (- 4) = \sqrt{25 (7)}$

Step 2.2.7.2

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

f (- 4) = \sqrt{5^{2} \cdot 7}

$f (- 4) = \sqrt{5^{2} \cdot 7}$

f (- 4) = \sqrt{5^{2} \cdot 7}

$f (- 4) = \sqrt{5^{2} \cdot 7}$

Step 2.2.8

Pull terms out from under the radical.

f (- 4) = 5 \sqrt{7}

$f (- 4) = 5 \sqrt{7}$

Step 2.2.9

The final answer is

5 \sqrt{7}

$5 \sqrt{7}$ .

5 \sqrt{7}

$5 \sqrt{7}$

5 \sqrt{7}

$5 \sqrt{7}$

Step 2.3

The

y

$y$ value at

x = - 4

$x = - 4$ is

5 \sqrt{7}

$5 \sqrt{7}$ .

y = 5 \sqrt{7}

$y = 5 \sqrt{7}$

y = 5 \sqrt{7}

$y = 5 \sqrt{7}$

Step 3

Find the

y

$y$ value at

x = - 5

$x = - 5$ .

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Step 3.1

Replace the variable

x

$x$ with

- 5

$- 5$ in the expression.

f (- 5) = \sqrt{({(- 5)}^{2} + 9) ((- 5) + 3) ((- 5) - 3)}

$f (- 5) = \sqrt{({(- 5)}^{2} + 9) ((- 5) + 3) ((- 5) - 3)}$

Step 3.2

Simplify the result.

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Step 3.2.1

Raise

- 5

$- 5$ to the power of

2

$2$ .

f (- 5) = \sqrt{(25 + 9) (- 5 + 3) (- 5 - 3)}

$f (- 5) = \sqrt{(25 + 9) (- 5 + 3) (- 5 - 3)}$

Step 3.2.2

Add

25

$25$ and

9

$9$ .

f (- 5) = \sqrt{34 (- 5 + 3) (- 5 - 3)}

$f (- 5) = \sqrt{34 (- 5 + 3) (- 5 - 3)}$

Step 3.2.3

Add

- 5

$- 5$ and

3

$3$ .

f (- 5) = \sqrt{34 \cdot (- 2 (- 5 - 3))}

$f (- 5) = \sqrt{34 \cdot (- 2 (- 5 - 3))}$

Step 3.2.4

Multiply

34

$34$ by

- 2

$- 2$ .

f (- 5) = \sqrt{- 68 (- 5 - 3)}

$f (- 5) = \sqrt{- 68 (- 5 - 3)}$

Step 3.2.5

Subtract

3

$3$ from

- 5

$- 5$ .

f (- 5) = \sqrt{- 68 \cdot - 8}

$f (- 5) = \sqrt{- 68 \cdot - 8}$

Step 3.2.6

Multiply

- 68

$- 68$ by

- 8

$- 8$ .

f (- 5) = \sqrt{544}

$f (- 5) = \sqrt{544}$

Step 3.2.7

Rewrite

544

$544$ as

4^{2} \cdot 34

$4^{2} \cdot 34$ .

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Step 3.2.7.1

Factor

16

$16$ out of

544

$544$ .

f (- 5) = \sqrt{16 (34)}

$f (- 5) = \sqrt{16 (34)}$

Step 3.2.7.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

f (- 5) = \sqrt{4^{2} \cdot 34}

$f (- 5) = \sqrt{4^{2} \cdot 34}$

f (- 5) = \sqrt{4^{2} \cdot 34}

$f (- 5) = \sqrt{4^{2} \cdot 34}$

Step 3.2.8

Pull terms out from under the radical.

f (- 5) = 4 \sqrt{34}

$f (- 5) = 4 \sqrt{34}$

Step 3.2.9

The final answer is

4 \sqrt{34}

$4 \sqrt{34}$ .

4 \sqrt{34}

$4 \sqrt{34}$

4 \sqrt{34}

$4 \sqrt{34}$

Step 3.3

The

y

$y$ value at

x = - 5

$x = - 5$ is

4 \sqrt{34}

$4 \sqrt{34}$ .

y = 4 \sqrt{34}

$y = 4 \sqrt{34}$

y = 4 \sqrt{34}

$y = 4 \sqrt{34}$

Step 4

Find the

y

$y$ value at

x = 3

$x = 3$ .

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Step 4.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = \sqrt{({(3)}^{2} + 9) ((3) + 3) ((3) - 3)}

$f (3) = \sqrt{({(3)}^{2} + 9) ((3) + 3) ((3) - 3)}$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise

3

$3$ to the power of

2

$2$ .

f (3) = \sqrt{(9 + 9) (3 + 3) (3 - 3)}

$f (3) = \sqrt{(9 + 9) (3 + 3) (3 - 3)}$

Step 4.2.2

Add

9

$9$ and

9

$9$ .

f (3) = \sqrt{18 (3 + 3) (3 - 3)}

$f (3) = \sqrt{18 (3 + 3) (3 - 3)}$

Step 4.2.3

Add

3

$3$ and

3

$3$ .

f (3) = \sqrt{18 \cdot (6 (3 - 3))}

$f (3) = \sqrt{18 \cdot (6 (3 - 3))}$

Step 4.2.4

Multiply

18

$18$ by

6

$6$ .

f (3) = \sqrt{108 (3 - 3)}

$f (3) = \sqrt{108 (3 - 3)}$

Step 4.2.5

Subtract

3

$3$ from

3

$3$ .

f (3) = \sqrt{108 \cdot 0}

$f (3) = \sqrt{108 \cdot 0}$

Step 4.2.6

Multiply

108

$108$ by

0

$0$ .

f (3) = \sqrt{0}

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

Step 4.2.7

Rewrite

0

$0$ as

0^{2}

$0^{2}$ .

f (3) = \sqrt{0^{2}}

$f (3) = \sqrt{0^{2}}$

Step 4.2.8

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

f (3) = 0

$f (3) = 0$

Step 4.2.9

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 4.3

The

y

$y$ value at

x = 3

$x = 3$ is

0

$0$ .

y = 0

$y = 0$

y = 0

$y = 0$

Step 5

Find the

y

$y$ value at

x = 4

$x = 4$ .

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Step 5.1

Replace the variable

x

$x$ with

4

$4$ in the expression.

f (4) = \sqrt{({(4)}^{2} + 9) ((4) + 3) ((4) - 3)}

$f (4) = \sqrt{({(4)}^{2} + 9) ((4) + 3) ((4) - 3)}$

Step 5.2

Simplify the result.

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Step 5.2.1

Raise

4

$4$ to the power of

2

$2$ .

f (4) = \sqrt{(16 + 9) (4 + 3) (4 - 3)}

$f (4) = \sqrt{(16 + 9) (4 + 3) (4 - 3)}$

Step 5.2.2

Add

16

$16$ and

9

$9$ .

f (4) = \sqrt{25 (4 + 3) (4 - 3)}

$f (4) = \sqrt{25 (4 + 3) (4 - 3)}$

Step 5.2.3

Add

4

$4$ and

3

$3$ .

f (4) = \sqrt{25 \cdot (7 (4 - 3))}

$f (4) = \sqrt{25 \cdot (7 (4 - 3))}$

Step 5.2.4

Multiply

25

$25$ by

7

$7$ .

f (4) = \sqrt{175 (4 - 3)}

$f (4) = \sqrt{175 (4 - 3)}$

Step 5.2.5

Subtract

3

$3$ from

4

$4$ .

f (4) = \sqrt{175 \cdot 1}

$f (4) = \sqrt{175 \cdot 1}$

Step 5.2.6

Multiply

175

$175$ by

1

$1$ .

f (4) = \sqrt{175}

$f (4) = \sqrt{175}$

Step 5.2.7

Rewrite

175

$175$ as

5^{2} \cdot 7

$5^{2} \cdot 7$ .

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Step 5.2.7.1

Factor

25

$25$ out of

175

$175$ .

f (4) = \sqrt{25 (7)}

$f (4) = \sqrt{25 (7)}$

Step 5.2.7.2

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

f (4) = \sqrt{5^{2} \cdot 7}

$f (4) = \sqrt{5^{2} \cdot 7}$

f (4) = \sqrt{5^{2} \cdot 7}

$f (4) = \sqrt{5^{2} \cdot 7}$

Step 5.2.8

Pull terms out from under the radical.

f (4) = 5 \sqrt{7}

$f (4) = 5 \sqrt{7}$

Step 5.2.9

The final answer is

5 \sqrt{7}

$5 \sqrt{7}$ .

5 \sqrt{7}

$5 \sqrt{7}$

Step 5.3

The

$y$ value at

$x = 4$ is

$5 \sqrt{7}$ .

$y = 5 \sqrt{7}$

Step 6

Find the

$y$ value at

$x = 5$ .

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Step 6.1

Replace the variable

$x$ with

$5$ in the expression.

$f (5) = \sqrt{({(5)}^{2} + 9) ((5) + 3) ((5) - 3)}$

Step 6.2

Simplify the result.

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Step 6.2.1

Raise

$5$ to the power of

$2$ .

$f (5) = \sqrt{(25 + 9) (5 + 3) (5 - 3)}$

Step 6.2.2

Add

$25$ and

$9$ .

$f (5) = \sqrt{34 (5 + 3) (5 - 3)}$

Step 6.2.3

Add

$5$ and

$3$ .

$f (5) = \sqrt{34 \cdot (8 (5 - 3))}$

Step 6.2.4

Multiply

$34$ by

$8$ .

$f (5) = \sqrt{272 (5 - 3)}$

Step 6.2.5

Subtract

$3$ from

$5$ .

$f (5) = \sqrt{272 \cdot 2}$

Step 6.2.6

Multiply

$272$ by

$2$ .

$f (5) = \sqrt{544}$

Step 6.2.7

Rewrite

$544$ as

$4^{2} \cdot 34$ .

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Step 6.2.7.1

Factor

$16$ out of

$544$ .

$f (5) = \sqrt{16 (34)}$

Step 6.2.7.2

Rewrite

$16$ as

$4^{2}$ .

$f (5) = \sqrt{4^{2} \cdot 34}$

Step 6.2.8

Pull terms out from under the radical.

$f (5) = 4 \sqrt{34}$

Step 6.2.9

The final answer is

$4 \sqrt{34}$ .

$4 \sqrt{34}$

Step 6.3

The

$y$ value at

$x = 5$ is

$4 \sqrt{34}$ .

$y = 4 \sqrt{34}$

Step 7

List the points to graph.

$(- 3, 0), (- 4, 13.22875655), (- 5, 23.32380757), (3, 0), (4, 13.22875655), (5, 23.32380757)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 5 & 23.324 \\ - 4 & 13.229 \\ - 3 & 0 \\ 3 & 0 \\ 4 & 13.229 \\ 5 & 23.324 \end{array}$

Step 9