Basic Math Examples

$\frac{6 h^{4} - 5}{8 h^{5}} - \frac{5 h}{8 h^{5} - 4}$

Step 1

Factor $4$ out of $8 h^{5} - 4$ .

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Step 1.1

Factor $4$ out of $8 h^{5}$ .

$\frac{6 h^{4} - 5}{8 h^{5}} - \frac{5 h}{4 (2 h^{5}) - 4}$

Step 1.2

Factor $4$ out of $- 4$ .

$\frac{6 h^{4} - 5}{8 h^{5}} - \frac{5 h}{4 (2 h^{5}) + 4 (- 1)}$

Step 1.3

Factor $4$ out of $4 (2 h^{5}) + 4 (- 1)$ .

$\frac{6 h^{4} - 5}{8 h^{5}} - \frac{5 h}{4 (2 h^{5} - 1)}$

Step 2

To write $\frac{6 h^{4} - 5}{8 h^{5}}$ as a fraction with a common denominator, multiply by $\frac{2 h^{5} - 1}{2 h^{5} - 1}$ .

$\frac{6 h^{4} - 5}{8 h^{5}} \cdot \frac{2 h^{5} - 1}{2 h^{5} - 1} - \frac{5 h}{4 (2 h^{5} - 1)}$

Step 3

To write $- \frac{5 h}{4 (2 h^{5} - 1)}$ as a fraction with a common denominator, multiply by $\frac{2 h^{5}}{2 h^{5}}$ .

$\frac{6 h^{4} - 5}{8 h^{5}} \cdot \frac{2 h^{5} - 1}{2 h^{5} - 1} - \frac{5 h}{4 (2 h^{5} - 1)} \cdot \frac{2 h^{5}}{2 h^{5}}$

Step 4

Write each expression with a common denominator of $8 (2 h^{5} - 1) h^{5}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{6 h^{4} - 5}{8 h^{5}}$ by $\frac{2 h^{5} - 1}{2 h^{5} - 1}$ .

$\frac{(6 h^{4} - 5) (2 h^{5} - 1)}{8 h^{5} (2 h^{5} - 1)} - \frac{5 h}{4 (2 h^{5} - 1)} \cdot \frac{2 h^{5}}{2 h^{5}}$

Step 4.2

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

$\frac{(6 h^{4} - 5) (2 h^{5} - 1)}{8 h^{5} (2 h^{5} - 1)} - \frac{5 h (2 h^{5})}{4 (2 h^{5} - 1) (2 h^{5})}$

Step 4.3

Multiply $2$ by $4$ .

$\frac{(6 h^{4} - 5) (2 h^{5} - 1)}{8 h^{5} (2 h^{5} - 1)} - \frac{5 h (2 h^{5})}{8 (2 h^{5} - 1) h^{5}}$

Step 4.4

Reorder the factors of $8 (2 h^{5} - 1) h^{5}$ .

$\frac{(6 h^{4} - 5) (2 h^{5} - 1)}{8 h^{5} (2 h^{5} - 1)} - \frac{5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(6 h^{4} - 5) (2 h^{5} - 1) - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(6 h^{4} - 5) (2 h^{5} - 1)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{6 h^{4} (2 h^{5} - 1) - 5 (2 h^{5} - 1) - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.1.2

Apply the distributive property.

$\frac{6 h^{4} (2 h^{5}) + 6 h^{4} \cdot - 1 - 5 (2 h^{5} - 1) - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.1.3

Apply the distributive property.

$\frac{6 h^{4} (2 h^{5}) + 6 h^{4} \cdot - 1 - 5 (2 h^{5}) - 5 \cdot - 1 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.2

Simplify each term.

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Step 6.2.1

Rewrite using the commutative property of multiplication.

$\frac{6 \cdot 2 h^{4} h^{5} + 6 h^{4} \cdot - 1 - 5 (2 h^{5}) - 5 \cdot - 1 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.2.2

Multiply $h^{4}$ by $h^{5}$ by adding the exponents.

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Step 6.2.2.1

Move $h^{5}$ .

$\frac{6 \cdot 2 (h^{5} h^{4}) + 6 h^{4} \cdot - 1 - 5 (2 h^{5}) - 5 \cdot - 1 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.2.2.2

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

$\frac{6 \cdot 2 h^{5 + 4} + 6 h^{4} \cdot - 1 - 5 (2 h^{5}) - 5 \cdot - 1 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.2.2.3

Add $5$ and $4$ .

$\frac{6 \cdot 2 h^{9} + 6 h^{4} \cdot - 1 - 5 (2 h^{5}) - 5 \cdot - 1 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.2.3

Multiply $6$ by $2$ .

$\frac{12 h^{9} + 6 h^{4} \cdot - 1 - 5 (2 h^{5}) - 5 \cdot - 1 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.2.4

Multiply $- 1$ by $6$ .

$\frac{12 h^{9} - 6 h^{4} - 5 (2 h^{5}) - 5 \cdot - 1 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.2.5

Multiply $2$ by $- 5$ .

$\frac{12 h^{9} - 6 h^{4} - 10 h^{5} - 5 \cdot - 1 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.2.6

Multiply $- 5$ by $- 1$ .

$\frac{12 h^{9} - 6 h^{4} - 10 h^{5} + 5 - 5 h (2 h^{5})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.3

Rewrite using the commutative property of multiplication.

$\frac{12 h^{9} - 6 h^{4} - 10 h^{5} + 5 - 5 \cdot 2 h \cdot h^{5}}{8 h^{5} (2 h^{5} - 1)}$

Step 6.4

Multiply $h$ by $h^{5}$ by adding the exponents.

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Step 6.4.1

Move $h^{5}$ .

$\frac{12 h^{9} - 6 h^{4} - 10 h^{5} + 5 - 5 \cdot 2 (h^{5} h)}{8 h^{5} (2 h^{5} - 1)}$

Step 6.4.2

Multiply $h^{5}$ by $h$ .

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Step 6.4.2.1

Raise $h$ to the power of $1$ .

$\frac{12 h^{9} - 6 h^{4} - 10 h^{5} + 5 - 5 \cdot 2 (h^{5} h^{1})}{8 h^{5} (2 h^{5} - 1)}$

Step 6.4.2.2

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

$\frac{12 h^{9} - 6 h^{4} - 10 h^{5} + 5 - 5 \cdot 2 h^{5 + 1}}{8 h^{5} (2 h^{5} - 1)}$

Step 6.4.3

Add $5$ and $1$ .

$\frac{12 h^{9} - 6 h^{4} - 10 h^{5} + 5 - 5 \cdot 2 h^{6}}{8 h^{5} (2 h^{5} - 1)}$

Step 6.5

Multiply $- 5$ by $2$ .

$\frac{12 h^{9} - 6 h^{4} - 10 h^{5} + 5 - 10 h^{6}}{8 h^{5} (2 h^{5} - 1)}$

Step 6.6

Reorder terms.

$\frac{12 h^{9} - 10 h^{6} - 10 h^{5} - 6 h^{4} + 5}{8 h^{5} (2 h^{5} - 1)}$