Calculus Examples

$\frac{60 x^{\frac{5}{2}} + 18 x^{\frac{7}{2}}}{16 x^{6}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $6 x^{\frac{5}{2}}$ out of $60 x^{\frac{5}{2}} + 18 x^{\frac{7}{2}}$ .

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Step 1.1.1

Factor $6 x^{\frac{5}{2}}$ out of $60 x^{\frac{5}{2}}$ .

$\frac{6 x^{\frac{5}{2}} (10) + 18 x^{\frac{7}{2}}}{16 x^{6}}$

Step 1.1.2

Factor $6 x^{\frac{5}{2}}$ out of $18 x^{\frac{7}{2}}$ .

$\frac{6 x^{\frac{5}{2}} (10) + 6 x^{\frac{5}{2}} (3 x^{\frac{2}{2}})}{16 x^{6}}$

Step 1.1.3

Factor $6 x^{\frac{5}{2}}$ out of $6 x^{\frac{5}{2}} (10) + 6 x^{\frac{5}{2}} (3 x^{\frac{2}{2}})$ .

$\frac{6 x^{\frac{5}{2}} (10 + 3 x^{\frac{2}{2}})}{16 x^{6}}$

Step 1.2

Divide $2$ by $2$ .

$\frac{6 x^{\frac{5}{2}} (10 + 3 x^{1})}{16 x^{6}}$

Step 1.3

Simplify.

$\frac{6 x^{\frac{5}{2}} (10 + 3 x)}{16 x^{6}}$

Step 2

Move $x^{\frac{5}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{6 (10 + 3 x)}{16 x^{6} x^{- \frac{5}{2}}}$

Step 3

Multiply $x^{6}$ by $x^{- \frac{5}{2}}$ by adding the exponents.

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Step 3.1

Move $x^{- \frac{5}{2}}$ .

$\frac{6 (10 + 3 x)}{16 (x^{- \frac{5}{2}} x^{6})}$

Step 3.2

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

$\frac{6 (10 + 3 x)}{16 x^{- \frac{5}{2} + 6}}$

Step 3.3

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{6 (10 + 3 x)}{16 x^{- \frac{5}{2} + 6 \cdot \frac{2}{2}}}$

Step 3.4

Combine $6$ and $\frac{2}{2}$ .

$\frac{6 (10 + 3 x)}{16 x^{- \frac{5}{2} + \frac{6 \cdot 2}{2}}}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{6 (10 + 3 x)}{16 x^{\frac{- 5 + 6 \cdot 2}{2}}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $6$ by $2$ .

$\frac{6 (10 + 3 x)}{16 x^{\frac{- 5 + 12}{2}}}$

Step 3.6.2

Add $- 5$ and $12$ .

$\frac{6 (10 + 3 x)}{16 x^{\frac{7}{2}}}$

Step 4

Factor $2$ out of $6 (10 + 3 x)$ .

$\frac{2 (3 (10 + 3 x))}{16 x^{\frac{7}{2}}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $2$ out of $16 x^{\frac{7}{2}}$ .

$\frac{2 (3 (10 + 3 x))}{2 (8 x^{\frac{7}{2}})}$

Step 5.2

Cancel the common factor.

$\frac{2 (3 (10 + 3 x))}{2 (8 x^{\frac{7}{2}})}$

Step 5.3

Rewrite the expression.

$\frac{3 (10 + 3 x)}{8 x^{\frac{7}{2}}}$