Algebra Examples

$h (x) = \frac{8}{\sqrt{8 x + 10}}$

Step 1

Factor $2$ out of $8 x + 10$ .

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Step 1.1

Factor $2$ out of $8 x$ .

$h (x) = \frac{8}{\sqrt{2 (4 x) + 10}}$

Step 1.2

Factor $2$ out of $10$ .

$h (x) = \frac{8}{\sqrt{2 (4 x) + 2 (5)}}$

Step 1.3

Factor $2$ out of $2 (4 x) + 2 (5)$ .

$h (x) = \frac{8}{\sqrt{2 (4 x + 5)}}$

Step 2

Multiply $\frac{8}{\sqrt{2 (4 x + 5)}}$ by $\frac{\sqrt{2 (4 x + 5)}}{\sqrt{2 (4 x + 5)}}$ .

$h (x) = \frac{8}{\sqrt{2 (4 x + 5)}} \cdot \frac{\sqrt{2 (4 x + 5)}}{\sqrt{2 (4 x + 5)}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{8}{\sqrt{2 (4 x + 5)}}$ by $\frac{\sqrt{2 (4 x + 5)}}{\sqrt{2 (4 x + 5)}}$ .

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{\sqrt{2 (4 x + 5)} \sqrt{2 (4 x + 5)}}$

Step 3.2

Raise $\sqrt{2 (4 x + 5)}$ to the power of $1$ .

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{\sqrt{2 (4 x + 5)} \sqrt{2 (4 x + 5)}}$

Step 3.3

Raise $\sqrt{2 (4 x + 5)}$ to the power of $1$ .

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{\sqrt{2 (4 x + 5)} \sqrt{2 (4 x + 5)}}$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{{\sqrt{2 (4 x + 5)}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{{\sqrt{2 (4 x + 5)}}^{2}}$

Step 3.6

Rewrite ${\sqrt{2 (4 x + 5)}}^{2}$ as $2 (4 x + 5)$ .

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (4 x + 5)}$ as ${(2 (4 x + 5))}^{\frac{1}{2}}$ .

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{{({(2 (4 x + 5))}^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{{(2 (4 x + 5))}^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{{(2 (4 x + 5))}^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{{(2 (4 x + 5))}^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{2 (4 x + 5)}$

Step 3.6.5

Simplify.

$h (x) = \frac{8 \sqrt{2 (4 x + 5)}}{2 (4 x + 5)}$

Step 4

Cancel the common factor of $8$ and $2$ .

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Step 4.1

Factor $2$ out of $8 \sqrt{2 (4 x + 5)}$ .

$h (x) = \frac{2 (4 \sqrt{2 (4 x + 5)})}{2 (4 x + 5)}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Cancel the common factor.

$h (x) = \frac{2 (4 \sqrt{2 (4 x + 5)})}{2 (4 x + 5)}$

Step 4.2.2

Rewrite the expression.

$h (x) = \frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5}$

Step 5

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 6

Find the components of the definition.

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Step 6.1

Evaluate the function at $x = x + h$ .

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Step 6.1.1

Replace the variable $x$ with $x + h$ in the expression.

$h (x + h) = \frac{4 \sqrt{2 (4 (x + h) + 5)}}{4 (x + h) + 5}$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Apply the distributive property.

$h (x + h) = \frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 (x + h) + 5}$

Step 6.1.2.2

Apply the distributive property.

$h (x + h) = \frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5}$

Step 6.1.2.3

The final answer is $\frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5}$ .

$\frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5}$

Step 6.2

Find the components of the definition.

$h (x + h) = \frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5}$

$h (x) = \frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5}$

$h (x + h) = \frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5}$

$h (x) = \frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5}$

Step 7

Plug in the components.

$\frac{h (x + h) - h x}{h} = \frac{\frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5} - (\frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5})}{h}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

To write $\frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5}$ as a fraction with a common denominator, multiply by $\frac{4 x + 5}{4 x + 5}$ .

$\frac{\frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5} \cdot \frac{4 x + 5}{4 x + 5} - \frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5}}{h}$

Step 8.1.2

To write $- \frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5}$ as a fraction with a common denominator, multiply by $\frac{4 x + 4 h + 5}{4 x + 4 h + 5}$ .

$\frac{\frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5} \cdot \frac{4 x + 5}{4 x + 5} - \frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5} \cdot \frac{4 x + 4 h + 5}{4 x + 4 h + 5}}{h}$

Step 8.1.3

Write each expression with a common denominator of $(4 x + 4 h + 5) (4 x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1.3.1

Multiply $\frac{4 \sqrt{2 (4 x + 4 h + 5)}}{4 x + 4 h + 5}$ by $\frac{4 x + 5}{4 x + 5}$ .

$\frac{\frac{4 \sqrt{2 (4 x + 4 h + 5)} (4 x + 5)}{(4 x + 4 h + 5) (4 x + 5)} - \frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5} \cdot \frac{4 x + 4 h + 5}{4 x + 4 h + 5}}{h}$

Step 8.1.3.2

Multiply $\frac{4 \sqrt{2 (4 x + 5)}}{4 x + 5}$ by $\frac{4 x + 4 h + 5}{4 x + 4 h + 5}$ .

$\frac{\frac{4 \sqrt{2 (4 x + 4 h + 5)} (4 x + 5)}{(4 x + 4 h + 5) (4 x + 5)} - \frac{4 \sqrt{2 (4 x + 5)} (4 x + 4 h + 5)}{(4 x + 5) (4 x + 4 h + 5)}}{h}$

Step 8.1.3.3

Reorder the factors of $(4 x + 5) (4 x + 4 h + 5)$ .

$\frac{\frac{4 \sqrt{2 (4 x + 4 h + 5)} (4 x + 5)}{(4 x + 4 h + 5) (4 x + 5)} - \frac{4 \sqrt{2 (4 x + 5)} (4 x + 4 h + 5)}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.4

Combine the numerators over the common denominator.

$\frac{\frac{4 \sqrt{2 (4 x + 4 h + 5)} (4 x + 5) - 4 \sqrt{2 (4 x + 5)} (4 x + 4 h + 5)}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5

Rewrite $\frac{4 \sqrt{2 (4 x + 4 h + 5)} (4 x + 5) - 4 \sqrt{2 (4 x + 5)} (4 x + 4 h + 5)}{(4 x + 4 h + 5) (4 x + 5)}$ in a factored form.

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Step 8.1.5.1

Factor $4$ out of $4 \sqrt{2 (4 x + 4 h + 5)} (4 x + 5) - 4 \sqrt{2 (4 x + 5)} (4 x + 4 h + 5)$ .

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Step 8.1.5.1.1

Factor $4$ out of $4 \sqrt{2 (4 x + 4 h + 5)} (4 x + 5)$ .

$\frac{\frac{4 (\sqrt{2 (4 x + 4 h + 5)} (4 x + 5)) - 4 \sqrt{2 (4 x + 5)} (4 x + 4 h + 5)}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.1.2

Factor $4$ out of $- 4 \sqrt{2 (4 x + 5)} (4 x + 4 h + 5)$ .

$\frac{\frac{4 (\sqrt{2 (4 x + 4 h + 5)} (4 x + 5)) + 4 (- \sqrt{2 (4 x + 5)} (4 x + 4 h + 5))}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.1.3

Factor $4$ out of $4 (\sqrt{2 (4 x + 4 h + 5)} (4 x + 5)) + 4 (- \sqrt{2 (4 x + 5)} (4 x + 4 h + 5))$ .

$\frac{\frac{4 (\sqrt{2 (4 x + 4 h + 5)} (4 x + 5) - \sqrt{2 (4 x + 5)} (4 x + 4 h + 5))}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.2

Apply the distributive property.

$\frac{\frac{4 (\sqrt{2 (4 x + 4 h + 5)} (4 x) + \sqrt{2 (4 x + 4 h + 5)} \cdot 5 - \sqrt{2 (4 x + 5)} (4 x + 4 h + 5))}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + \sqrt{2 (4 x + 4 h + 5)} \cdot 5 - \sqrt{2 (4 x + 5)} (4 x + 4 h + 5))}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.4

Move $5$ to the left of $\sqrt{2 (4 x + 4 h + 5)}$ .

$\frac{\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \cdot \sqrt{2 (4 x + 4 h + 5)} - \sqrt{2 (4 x + 5)} (4 x + 4 h + 5))}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.5

Apply the distributive property.

$\frac{\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - \sqrt{2 (4 x + 5)} (4 x) - \sqrt{2 (4 x + 5)} (4 h) - \sqrt{2 (4 x + 5)} \cdot 5)}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.6

Simplify.

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Step 8.1.5.6.1

Multiply $4$ by $- 1$ .

$\frac{\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - 4 \sqrt{2 (4 x + 5)} x - \sqrt{2 (4 x + 5)} (4 h) - \sqrt{2 (4 x + 5)} \cdot 5)}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.6.2

Multiply $4$ by $- 1$ .

$\frac{\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - 4 \sqrt{2 (4 x + 5)} x - 4 \sqrt{2 (4 x + 5)} h - \sqrt{2 (4 x + 5)} \cdot 5)}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.1.5.6.3

Multiply $5$ by $- 1$ .

$\frac{\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - 4 \sqrt{2 (4 x + 5)} x - 4 \sqrt{2 (4 x + 5)} h - 5 \sqrt{2 (4 x + 5)})}{(4 x + 4 h + 5) (4 x + 5)}}{h}$

Step 8.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - 4 \sqrt{2 (4 x + 5)} x - 4 \sqrt{2 (4 x + 5)} h - 5 \sqrt{2 (4 x + 5)})}{(4 x + 4 h + 5) (4 x + 5)} \cdot \frac{1}{h}$

Step 8.3

Multiply $\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - 4 \sqrt{2 (4 x + 5)} x - 4 \sqrt{2 (4 x + 5)} h - 5 \sqrt{2 (4 x + 5)})}{(4 x + 4 h + 5) (4 x + 5)}$ by $\frac{1}{h}$ .

$\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - 4 \sqrt{2 (4 x + 5)} x - 4 \sqrt{2 (4 x + 5)} h - 5 \sqrt{2 (4 x + 5)})}{(4 x + 4 h + 5) (4 x + 5) h}$

Step 8.4

Reorder factors in $\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - 4 \sqrt{2 (4 x + 5)} x - 4 \sqrt{2 (4 x + 5)} h - 5 \sqrt{2 (4 x + 5)})}{(4 x + 4 h + 5) (4 x + 5) h}$ .

$\frac{4 (4 \sqrt{2 (4 x + 4 h + 5)} x + 5 \sqrt{2 (4 x + 4 h + 5)} - 4 \sqrt{2 (4 x + 5)} x - 4 \sqrt{2 (4 x + 5)} h - 5 \sqrt{2 (4 x + 5)})}{h (4 x + 4 h + 5) (4 x + 5)}$

Step 9