Trigonometry Examples

$\frac{3 b}{\sqrt{3 b + 57 - 1}}$

Step 1

Set the radicand in $\sqrt{3 b + 57 - 1}$ greater than or equal to $0$ to find where the expression is defined.

$3 b + 57 - 1 \geq 0$

Step 2

Solve for $b$ .

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Step 2.1

Subtract $1$ from $57$ .

$3 b + 56 \geq 0$

Step 2.2

Subtract $56$ from both sides of the inequality.

$3 b \geq - 56$

Step 2.3

Divide each term in $3 b \geq - 56$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 b \geq - 56$ by $3$ .

$\frac{3 b}{3} \geq \frac{- 56}{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 b}{3} \geq \frac{- 56}{3}$

Step 2.3.2.1.2

Divide $b$ by $1$ .

$b \geq \frac{- 56}{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$b \geq - \frac{56}{3}$

Step 3

Set the denominator in $\frac{3 b}{\sqrt{3 b + 57 - 1}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{3 b + 57 - 1} = 0$

Step 4

Solve for $b$ .

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Step 4.1

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

${\sqrt{3 b + 57 - 1}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 b + 57 - 1}$ as ${(3 b + 57 - 1)}^{\frac{1}{2}}$ .

${({(3 b + 57 - 1)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(3 b + 57 - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(3 b + 57 - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(3 b + 57 - 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(3 b + 57 - 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(3 b + 57 - 1)}^{1} = 0^{2}$

Step 4.2.2.1.2

Subtract $1$ from $57$ .

${(3 b + 56)}^{1} = 0^{2}$

Step 4.2.2.1.3

Simplify.

$3 b + 56 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$3 b + 56 = 0$

Step 4.3

Solve for $b$ .

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Step 4.3.1

Subtract $56$ from both sides of the equation.

$3 b = - 56$

Step 4.3.2

Divide each term in $3 b = - 56$ by $3$ and simplify.

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Step 4.3.2.1

Divide each term in $3 b = - 56$ by $3$ .

$\frac{3 b}{3} = \frac{- 56}{3}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $3$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{3 b}{3} = \frac{- 56}{3}$

Step 4.3.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 56}{3}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Move the negative in front of the fraction.

$b = - \frac{56}{3}$

Step 5

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

Interval Notation:

$(- \frac{56}{3}, \infty)$

Set-Builder Notation:

${b | b > - \frac{56}{3}}$

Step 6