Calculus Examples

$- \frac{1}{2 {(7 - x)}^{\frac{1}{2}}}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{1}{2 \sqrt{{(7 - x)}^{1}}}$

Step 1.2

Anything raised to $1$ is the base itself.

$- \frac{1}{2 \sqrt{7 - x}}$

Step 2

Set the radicand in $\sqrt{7 - x}$ greater than or equal to $0$ to find where the expression is defined.

$7 - x \geq 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $7$ from both sides of the inequality.

$- x \geq - 7$

Step 3.2

Divide each term in $- x \geq - 7$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x \geq - 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 7}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 7}{- 1}$

Step 3.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 7}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 7$ by $- 1$ .

$x \leq 7$

Step 4

Set the denominator in $\frac{1}{2 \sqrt{7 - x}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{7 - x} = 0$

Step 5

Solve for $x$ .

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Step 5.1

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

${(2 \sqrt{7 - x})}^{2} = 0^{2}$

Step 5.2

Simplify each side of the equation.

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Apply the product rule to $2 {(7 - x)}^{\frac{1}{2}}$ .

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

Step 5.2.2.1.2

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

$4 {({(7 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 5.2.2.1.3

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

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Step 5.2.2.1.3.1

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

$4 {(7 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 5.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.2.1.3.2.1

Cancel the common factor.

$4 {(7 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 5.2.2.1.3.2.2

Rewrite the expression.

$4 {(7 - x)}^{1} = 0^{2}$

Step 5.2.2.1.4

Simplify.

$4 (7 - x) = 0^{2}$

Step 5.2.2.1.5

Apply the distributive property.

$4 \cdot 7 + 4 (- x) = 0^{2}$

Step 5.2.2.1.6

Multiply.

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Step 5.2.2.1.6.1

Multiply $4$ by $7$ .

$28 + 4 (- x) = 0^{2}$

Step 5.2.2.1.6.2

Multiply $- 1$ by $4$ .

$28 - 4 x = 0^{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

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

$28 - 4 x = 0$

Step 5.3

Solve for $x$ .

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Step 5.3.1

Subtract $28$ from both sides of the equation.

$- 4 x = - 28$

Step 5.3.2

Divide each term in $- 4 x = - 28$ by $- 4$ and simplify.

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Step 5.3.2.1

Divide each term in $- 4 x = - 28$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 28}{- 4}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $- 4$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 28}{- 4}$

Step 5.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 28}{- 4}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $- 28$ by $- 4$ .

$x = 7$

Step 6

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

Interval Notation:

$(- \infty, 7)$

Set-Builder Notation:

${x | x < 7}$

Step 7