Basic Math Examples

${(a + 5 b)}^{7} \div {(a + 5 b)}^{3}$

Step 1

Rewrite the division as a fraction.

$\frac{{(a + 5 b)}^{7}}{{(a + 5 b)}^{3}}$

Step 2

Cancel the common factor of ${(a + 5 b)}^{7}$ and ${(a + 5 b)}^{3}$ .

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Step 2.1

Factor ${(a + 5 b)}^{3}$ out of ${(a + 5 b)}^{7}$ .

$\frac{{(a + 5 b)}^{3} {(a + 5 b)}^{4}}{{(a + 5 b)}^{3}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Multiply by $1$ .

$\frac{{(a + 5 b)}^{3} {(a + 5 b)}^{4}}{{(a + 5 b)}^{3} \cdot 1}$

Step 2.2.2

Cancel the common factor.

$\frac{{(a + 5 b)}^{3} {(a + 5 b)}^{4}}{{(a + 5 b)}^{3} \cdot 1}$

Step 2.2.3

Rewrite the expression.

$\frac{{(a + 5 b)}^{4}}{1}$

Step 2.2.4

Divide ${(a + 5 b)}^{4}$ by $1$ .

${(a + 5 b)}^{4}$

Step 3

Use the Binomial Theorem.

$a^{4} + 4 a^{3} (5 b) + 6 a^{2} {(5 b)}^{2} + 4 a {(5 b)}^{3} + {(5 b)}^{4}$

Step 4

Simplify each term.

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Step 4.1

Rewrite using the commutative property of multiplication.

$a^{4} + 4 \cdot 5 a^{3} b + 6 a^{2} {(5 b)}^{2} + 4 a {(5 b)}^{3} + {(5 b)}^{4}$

Step 4.2

Multiply $4$ by $5$ .

$a^{4} + 20 a^{3} b + 6 a^{2} {(5 b)}^{2} + 4 a {(5 b)}^{3} + {(5 b)}^{4}$

Step 4.3

Apply the product rule to $5 b$ .

$a^{4} + 20 a^{3} b + 6 a^{2} (5^{2} b^{2}) + 4 a {(5 b)}^{3} + {(5 b)}^{4}$

Step 4.4

Rewrite using the commutative property of multiplication.

$a^{4} + 20 a^{3} b + 6 \cdot 5^{2} a^{2} b^{2} + 4 a {(5 b)}^{3} + {(5 b)}^{4}$

Step 4.5

Raise $5$ to the power of $2$ .

$a^{4} + 20 a^{3} b + 6 \cdot 25 a^{2} b^{2} + 4 a {(5 b)}^{3} + {(5 b)}^{4}$

Step 4.6

Multiply $6$ by $25$ .

$a^{4} + 20 a^{3} b + 150 a^{2} b^{2} + 4 a {(5 b)}^{3} + {(5 b)}^{4}$

Step 4.7

Apply the product rule to $5 b$ .

$a^{4} + 20 a^{3} b + 150 a^{2} b^{2} + 4 a (5^{3} b^{3}) + {(5 b)}^{4}$

Step 4.8

Rewrite using the commutative property of multiplication.

$a^{4} + 20 a^{3} b + 150 a^{2} b^{2} + 4 \cdot 5^{3} a b^{3} + {(5 b)}^{4}$

Step 4.9

Raise $5$ to the power of $3$ .

$a^{4} + 20 a^{3} b + 150 a^{2} b^{2} + 4 \cdot 125 a b^{3} + {(5 b)}^{4}$

Step 4.10

Multiply $4$ by $125$ .

$a^{4} + 20 a^{3} b + 150 a^{2} b^{2} + 500 a b^{3} + {(5 b)}^{4}$

Step 4.11

Apply the product rule to $5 b$ .

$a^{4} + 20 a^{3} b + 150 a^{2} b^{2} + 500 a b^{3} + 5^{4} b^{4}$

Step 4.12

Raise $5$ to the power of $4$ .

$a^{4} + 20 a^{3} b + 150 a^{2} b^{2} + 500 a b^{3} + 625 b^{4}$