Algebra Examples

${(2 a - b)}^{2} \div \frac{4 a^{3} - a b^{2}}{3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

${(2 a - b)}^{2} \frac{3}{4 a^{3} - a b^{2}}$

Step 2

Rewrite ${(2 a - b)}^{2}$ as $(2 a - b) (2 a - b)$ .

$(2 a - b) (2 a - b) \frac{3}{4 a^{3} - a b^{2}}$

Step 3

Expand $(2 a - b) (2 a - b)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$(2 a (2 a - b) - b (2 a - b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 3.2

Apply the distributive property.

$(2 a (2 a) + 2 a (- b) - b (2 a - b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 3.3

Apply the distributive property.

$(2 a (2 a) + 2 a (- b) - b (2 a) - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$(2 \cdot 2 a \cdot a + 2 a (- b) - b (2 a) - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.2.1

Move $a$ .

$(2 \cdot 2 (a \cdot a) + 2 a (- b) - b (2 a) - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.2.2

Multiply $a$ by $a$ .

$(2 \cdot 2 a^{2} + 2 a (- b) - b (2 a) - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.3

Multiply $2$ by $2$ .

$(4 a^{2} + 2 a (- b) - b (2 a) - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$(4 a^{2} + 2 \cdot - 1 a b - b (2 a) - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.5

Multiply $2$ by $- 1$ .

$(4 a^{2} - 2 a b - b (2 a) - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.6

Rewrite using the commutative property of multiplication.

$(4 a^{2} - 2 a b - 1 \cdot 2 b a - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.7

Multiply $- 1$ by $2$ .

$(4 a^{2} - 2 a b - 2 b a - b (- b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.8

Rewrite using the commutative property of multiplication.

$(4 a^{2} - 2 a b - 2 b a - 1 \cdot - 1 b \cdot b) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.9

Multiply $b$ by $b$ by adding the exponents.

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Step 4.1.9.1

Move $b$ .

$(4 a^{2} - 2 a b - 2 b a - 1 \cdot - 1 (b \cdot b)) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.9.2

Multiply $b$ by $b$ .

$(4 a^{2} - 2 a b - 2 b a - 1 \cdot - 1 b^{2}) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.10

Multiply $- 1$ by $- 1$ .

$(4 a^{2} - 2 a b - 2 b a + 1 b^{2}) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.1.11

Multiply $b^{2}$ by $1$ .

$(4 a^{2} - 2 a b - 2 b a + b^{2}) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.2

Subtract $2 b a$ from $- 2 a b$ .

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Step 4.2.1

Move $b$ .

$(4 a^{2} - 2 a b - 2 a b + b^{2}) \frac{3}{4 a^{3} - a b^{2}}$

Step 4.2.2

Subtract $2 a b$ from $- 2 a b$ .

$(4 a^{2} - 4 a b + b^{2}) \frac{3}{4 a^{3} - a b^{2}}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $a$ out of $4 a^{3} - a b^{2}$ .

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Step 5.1.1

Factor $a$ out of $4 a^{3}$ .

$(4 a^{2} - 4 a b + b^{2}) \frac{3}{a (4 a^{2}) - a b^{2}}$

Step 5.1.2

Factor $a$ out of $- a b^{2}$ .

$(4 a^{2} - 4 a b + b^{2}) \frac{3}{a (4 a^{2}) + a (- 1 b^{2})}$

Step 5.1.3

Factor $a$ out of $a (4 a^{2}) + a (- 1 b^{2})$ .

$(4 a^{2} - 4 a b + b^{2}) \frac{3}{a (4 a^{2} - 1 b^{2})}$

Step 5.2

Rewrite $4 a^{2}$ as ${(2 a)}^{2}$ .

$(4 a^{2} - 4 a b + b^{2}) \frac{3}{a ({(2 a)}^{2} - 1 b^{2})}$

Step 5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a$ and $b = b$ .

$(4 a^{2} - 4 a b + b^{2}) \frac{3}{a (2 a + b) (2 a - b)}$

Step 6

Multiply $4 a^{2} - 4 a b + b^{2}$ by $\frac{3}{a (2 a + b) (2 a - b)}$ .

$\frac{(4 a^{2} - 4 a b + b^{2}) \cdot 3}{a (2 a + b) (2 a - b)}$

Step 7

Factor using the perfect square rule.

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Step 7.1

Rewrite $4 a^{2}$ as ${(2 a)}^{2}$ .

$\frac{({(2 a)}^{2} - 4 a b + b^{2}) \cdot 3}{a (2 a + b) (2 a - b)}$

Step 7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 a b = 2 \cdot (2 a) \cdot b$

Step 7.3

Rewrite the polynomial.

$\frac{({(2 a)}^{2} - 2 \cdot (2 a) \cdot b + b^{2}) \cdot 3}{a (2 a + b) (2 a - b)}$

Step 7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 a$ and $b = b$ .

$\frac{{(2 a - b)}^{2} \cdot 3}{a (2 a + b) (2 a - b)}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of ${(2 a - b)}^{2}$ and $2 a - b$ .

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Step 8.1.1

Factor $2 a - b$ out of ${(2 a - b)}^{2} \cdot 3$ .

$\frac{(2 a - b) ((2 a - b) \cdot 3)}{a (2 a + b) (2 a - b)}$

Step 8.1.2

Cancel the common factors.

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Step 8.1.2.1

Factor $2 a - b$ out of $a (2 a + b) (2 a - b)$ .

$\frac{(2 a - b) ((2 a - b) \cdot 3)}{(2 a - b) (a (2 a + b))}$

Step 8.1.2.2

Cancel the common factor.

$\frac{(2 a - b) ((2 a - b) \cdot 3)}{(2 a - b) (a (2 a + b))}$

Step 8.1.2.3

Rewrite the expression.

$\frac{(2 a - b) \cdot 3}{a (2 a + b)}$

Step 8.2

Move $3$ to the left of $2 a - b$ .

$\frac{3 (2 a - b)}{a (2 a + b)}$