Trigonometry Examples

${(\sqrt{x} - 9)}^{4} + 1$

Step 1

Simplify ${(\sqrt{x} - 9)}^{4} + 1$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Use the Binomial Theorem.

$y = {\sqrt{x}}^{4} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Rewrite ${\sqrt{x}}^{4}$ as $x^{2}$ .

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Step 1.1.2.1.1

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

$y = {(x^{\frac{1}{2}})}^{4} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.1.2

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

$y = x^{\frac{1}{2} \cdot 4} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.1.3

Combine $\frac{1}{2}$ and $4$ .

$y = x^{\frac{4}{2}} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.1.4

Cancel the common factor of $4$ and $2$ .

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Step 1.1.2.1.4.1

Factor $2$ out of $4$ .

$y = x^{\frac{2 \cdot 2}{2}} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.1.4.2

Cancel the common factors.

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Step 1.1.2.1.4.2.1

Factor $2$ out of $2$ .

$y = x^{\frac{2 \cdot 2}{2 (1)}} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.1.4.2.2

Cancel the common factor.

$y = x^{\frac{2 \cdot 2}{2 \cdot 1}} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.1.4.2.3

Rewrite the expression.

$y = x^{\frac{2}{1}} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.1.4.2.4

Divide $2$ by $1$ .

$y = x^{2} + 4 {\sqrt{x}}^{3} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.2

Rewrite ${\sqrt{x}}^{3}$ as $\sqrt{x^{3}}$ .

$y = x^{2} + 4 \sqrt{x^{3}} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.3

Factor out $x^{2}$ .

$y = x^{2} + 4 \sqrt{x^{2} x} \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.4

Pull terms out from under the radical.

$y = x^{2} + 4 (x \sqrt{x}) \cdot - 9 + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.5

Multiply $- 9$ by $4$ .

$y = x^{2} - 36 x \sqrt{x} + 6 {\sqrt{x}}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.1.2.6.1

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

$y = x^{2} - 36 x \sqrt{x} + 6 {(x^{\frac{1}{2}})}^{2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.6.2

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

$y = x^{2} - 36 x \sqrt{x} + 6 x^{\frac{1}{2} \cdot 2} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = x^{2} - 36 x \sqrt{x} + 6 x^{\frac{2}{2}} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.6.4

Cancel the common factor of $2$ .

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Step 1.1.2.6.4.1

Cancel the common factor.

$y = x^{2} - 36 x \sqrt{x} + 6 x^{\frac{2}{2}} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.6.4.2

Rewrite the expression.

$y = x^{2} - 36 x \sqrt{x} + 6 x^{1} {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.6.5

Simplify.

$y = x^{2} - 36 x \sqrt{x} + 6 x {(- 9)}^{2} + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.7

Raise $- 9$ to the power of $2$ .

$y = x^{2} - 36 x \sqrt{x} + 6 x \cdot 81 + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.8

Multiply $81$ by $6$ .

$y = x^{2} - 36 x \sqrt{x} + 486 x + 4 \sqrt{x} {(- 9)}^{3} + {(- 9)}^{4} + 1$

Step 1.1.2.9

Raise $- 9$ to the power of $3$ .

$y = x^{2} - 36 x \sqrt{x} + 486 x + 4 \sqrt{x} \cdot - 729 + {(- 9)}^{4} + 1$

Step 1.1.2.10

Multiply $- 729$ by $4$ .

$y = x^{2} - 36 x \sqrt{x} + 486 x - 2916 \sqrt{x} + {(- 9)}^{4} + 1$

Step 1.1.2.11

Raise $- 9$ to the power of $4$ .

$y = x^{2} - 36 x \sqrt{x} + 486 x - 2916 \sqrt{x} + 6561 + 1$

Step 1.2

Simplify the expression.

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Step 1.2.1

Add $6561$ and $1$ .

$y = x^{2} - 36 x \sqrt{x} + 486 x - 2916 \sqrt{x} + 6562$

Step 1.2.2

Move $- 36 x \sqrt{x}$ .

$y = x^{2} + 486 x - 36 x \sqrt{x} - 2916 \sqrt{x} + 6562$

Step 2

Find the domain for $y = {(\sqrt{x} - 9)}^{4} + 1$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 2.1

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

$x \geq 0$

Step 2.2

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 3

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = x^{2} + 486 x - 36 x \sqrt{x} - 2916 \sqrt{x} + 6562$ .

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Step 3.1

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

$f (0) = {(0)}^{2} + 486 (0) - 36 \cdot 0 \sqrt{0} - 2916 \sqrt{0} + 6562$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$f (0) = 0 + 486 (0) - 36 \cdot (0 \sqrt{0}) - 2916 \sqrt{0} + 6562$

Step 3.2.1.2

Multiply $486$ by $0$ .

$f (0) = 0 + 0 - 36 \cdot (0 \sqrt{0}) - 2916 \sqrt{0} + 6562$

Step 3.2.1.3

Multiply $- 36$ by $0$ .

$f (0) = 0 + 0 + 0 \sqrt{0} - 2916 \sqrt{0} + 6562$

Step 3.2.1.4

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

$f (0) = 0 + 0 + 0 \sqrt{0^{2}} - 2916 \sqrt{0} + 6562$

Step 3.2.1.5

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

$f (0) = 0 + 0 + 0 \cdot 0 - 2916 \sqrt{0} + 6562$

Step 3.2.1.6

Multiply $0$ by $0$ .

$f (0) = 0 + 0 + 0 - 2916 \sqrt{0} + 6562$

Step 3.2.1.7

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

$f (0) = 0 + 0 + 0 - 2916 \sqrt{0^{2}} + 6562$

Step 3.2.1.8

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

$f (0) = 0 + 0 + 0 - 2916 \cdot 0 + 6562$

Step 3.2.1.9

Multiply $- 2916$ by $0$ .

$f (0) = 0 + 0 + 0 + 0 + 6562$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 0 + 6562$

Step 3.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 0 + 6562$

Step 3.2.2.3

Add $0$ and $0$ .

$f (0) = 0 + 6562$

Step 3.2.2.4

Add $0$ and $6562$ .

$f (0) = 6562$

Step 3.2.3

The final answer is $6562$ .

$6562$

Step 4

The radical expression end point is $(0, 6562)$ .

$(0, 6562)$

Step 5

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

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Step 5.1

Substitute the $x$ value $1$ into $f (x) = x^{2} + 486 x - 36 x \sqrt{x} - 2916 \sqrt{x} + 6562$ . In this case, the point is $(1, 4097)$ .

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Step 5.1.1

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

$f (1) = {(1)}^{2} + 486 (1) - 36 \cdot 1 \sqrt{1} - 2916 \sqrt{1} + 6562$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

One to any power is one.

$f (1) = 1 + 486 (1) - 36 \cdot (1 \sqrt{1}) - 2916 \sqrt{1} + 6562$

Step 5.1.2.1.2

Multiply $486$ by $1$ .

$f (1) = 1 + 486 - 36 \cdot (1 \sqrt{1}) - 2916 \sqrt{1} + 6562$

Step 5.1.2.1.3

Multiply $- 36$ by $1$ .

$f (1) = 1 + 486 - 36 \sqrt{1} - 2916 \sqrt{1} + 6562$

Step 5.1.2.1.4

Any root of $1$ is $1$ .

$f (1) = 1 + 486 - 36 \cdot 1 - 2916 \sqrt{1} + 6562$

Step 5.1.2.1.5

Multiply $- 36$ by $1$ .

$f (1) = 1 + 486 - 36 - 2916 \sqrt{1} + 6562$

Step 5.1.2.1.6

Any root of $1$ is $1$ .

$f (1) = 1 + 486 - 36 - 2916 \cdot 1 + 6562$

Step 5.1.2.1.7

Multiply $- 2916$ by $1$ .

$f (1) = 1 + 486 - 36 - 2916 + 6562$

Step 5.1.2.2

Simplify by adding and subtracting.

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Step 5.1.2.2.1

Add $1$ and $486$ .

$f (1) = 487 - 36 - 2916 + 6562$

Step 5.1.2.2.2

Subtract $36$ from $487$ .

$f (1) = 451 - 2916 + 6562$

Step 5.1.2.2.3

Subtract $2916$ from $451$ .

$f (1) = - 2465 + 6562$

Step 5.1.2.2.4

Add $- 2465$ and $6562$ .

$f (1) = 4097$

Step 5.1.2.3

The final answer is $4097$ .

$y = 4097$

Step 5.2

Substitute the $x$ value $2$ into $f (x) = x^{2} + 486 x - 36 x \sqrt{x} - 2916 \sqrt{x} + 6562$ . In this case, the point is $(2, 7538 - 2988 \sqrt{2})$ .

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Step 5.2.1

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

$f (2) = {(2)}^{2} + 486 (2) - 36 \cdot 2 \sqrt{2} - 2916 \sqrt{2} + 6562$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

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

$f (2) = 4 + 486 (2) - 36 \cdot (2 \sqrt{2}) - 2916 \sqrt{2} + 6562$

Step 5.2.2.1.2

Multiply $486$ by $2$ .

$f (2) = 4 + 972 - 36 \cdot (2 \sqrt{2}) - 2916 \sqrt{2} + 6562$

Step 5.2.2.1.3

Multiply $- 36$ by $2$ .

$f (2) = 4 + 972 - 72 \sqrt{2} - 2916 \sqrt{2} + 6562$

Step 5.2.2.2

Simplify by adding terms.

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Step 5.2.2.2.1

Add $4$ and $972$ .

$f (2) = 976 - 72 \sqrt{2} - 2916 \sqrt{2} + 6562$

Step 5.2.2.2.2

Add $976$ and $6562$ .

$f (2) = 7538 - 72 \sqrt{2} - 2916 \sqrt{2}$

Step 5.2.2.2.3

Subtract $2916 \sqrt{2}$ from $- 72 \sqrt{2}$ .

$f (2) = 7538 - 2988 \sqrt{2}$

Step 5.2.2.3

The final answer is $7538 - 2988 \sqrt{2}$ .

$y = 7538 - 2988 \sqrt{2}$

Step 5.3

The square root can be graphed using the points around the vertex $(0, 6562), (1, 4097), (2, 3312.33)$

$\begin{array}{c} x & y \\ 0 & 6562 \\ 1 & 4097 \\ 2 & 3312.33 \end{array}$

Step 6