Basic Math Examples

$(3 a^{2} \sqrt{2 b a} - 5 a^{3} \sqrt{2 a b}) (5 a^{3} \sqrt{2 b a} + 3 a^{2} \sqrt{2 a b})$

Step 1

Expand $(3 a^{2} \sqrt{2 b a} - 5 a^{3} \sqrt{2 a b}) (5 a^{3} \sqrt{2 b a} + 3 a^{2} \sqrt{2 a b})$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

$3 a^{2} \sqrt{2 b a} (5 a^{3} \sqrt{2 b a} + 3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a} + 3 a^{2} \sqrt{2 a b})$

Step 1.2

Apply the distributive property.

$3 a^{2} \sqrt{2 b a} (5 a^{3} \sqrt{2 b a}) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a} + 3 a^{2} \sqrt{2 a b})$

Step 1.3

Apply the distributive property.

$3 a^{2} \sqrt{2 b a} (5 a^{3} \sqrt{2 b a}) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2

Simplify and combine like terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $a^{2}$ by $a^{3}$ by adding the exponents.

Tap for more steps...

Step 2.1.1.1

Move $a^{3}$ .

$3 (a^{3} a^{2}) \sqrt{2 b a} (5 \sqrt{2 b a}) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.1.2

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

$3 a^{3 + 2} \sqrt{2 b a} (5 \sqrt{2 b a}) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.1.3

Add $3$ and $2$ .

$3 a^{5} \sqrt{2 b a} (5 \sqrt{2 b a}) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.2

Multiply $3 a^{5} \sqrt{2 b a} (5 \sqrt{2 b a})$ .

Tap for more steps...

Step 2.1.2.1

Multiply $5$ by $3$ .

$15 a^{5} \sqrt{2 b a} \sqrt{2 b a} + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.2.2

Raise $\sqrt{2 b a}$ to the power of $1$ .

$15 a^{5} ({\sqrt{2 b a}}^{1} \sqrt{2 b a}) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.2.3

Raise $\sqrt{2 b a}$ to the power of $1$ .

$15 a^{5} ({\sqrt{2 b a}}^{1} {\sqrt{2 b a}}^{1}) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.2.4

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

$15 a^{5} {\sqrt{2 b a}}^{1 + 1} + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.2.5

Add $1$ and $1$ .

$15 a^{5} {\sqrt{2 b a}}^{2} + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.3

Rewrite ${\sqrt{2 b a}}^{2}$ as $2 b a$ .

Tap for more steps...

Step 2.1.3.1

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

$15 a^{5} {({(2 b a)}^{\frac{1}{2}})}^{2} + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.3.2

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

$15 a^{5} {(2 b a)}^{\frac{1}{2} \cdot 2} + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.3.3

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

$15 a^{5} {(2 b a)}^{\frac{2}{2}} + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.3.4.1

Cancel the common factor.

$15 a^{5} {(2 b a)}^{\frac{2}{2}} + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.3.4.2

Rewrite the expression.

$15 a^{5} {(2 b a)}^{1} + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.3.5

Simplify.

$15 a^{5} (2 b a) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.4

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

Tap for more steps...

Step 2.1.4.1

Move $a$ .

$15 (a \cdot a^{5}) (2 b) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.4.2

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

Tap for more steps...

Step 2.1.4.2.1

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

$15 (a^{1} a^{5}) (2 b) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.4.2.2

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

$15 a^{1 + 5} (2 b) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.4.3

Add $1$ and $5$ .

$15 a^{6} (2 b) + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$15 \cdot 2 a^{6} b + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.6

Multiply $15$ by $2$ .

$30 a^{6} b + 3 a^{2} \sqrt{2 b a} (3 a^{2} \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.7

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

Tap for more steps...

Step 2.1.7.1

Move $a^{2}$ .

$30 a^{6} b + 3 (a^{2} a^{2}) \sqrt{2 b a} (3 \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.7.2

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

$30 a^{6} b + 3 a^{2 + 2} \sqrt{2 b a} (3 \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.7.3

Add $2$ and $2$ .

$30 a^{6} b + 3 a^{4} \sqrt{2 b a} (3 \sqrt{2 a b}) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8

Multiply $3 a^{4} \sqrt{2 b a} (3 \sqrt{2 a b})$ .

Tap for more steps...

Step 2.1.8.1

Multiply $3$ by $3$ .

$30 a^{6} b + 9 a^{4} \sqrt{2 b a} \sqrt{2 a b} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.2

Combine using the product rule for radicals.

$30 a^{6} b + 9 a^{4} \sqrt{2 a b (2 b a)} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.3

Multiply $2$ by $2$ .

$30 a^{6} b + 9 a^{4} \sqrt{4 a b (b a)} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.4

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

$30 a^{6} b + 9 a^{4} \sqrt{4 (a^{1} a) b \cdot b} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.5

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

$30 a^{6} b + 9 a^{4} \sqrt{4 (a^{1} a^{1}) b \cdot b} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.6

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

$30 a^{6} b + 9 a^{4} \sqrt{4 a^{1 + 1} b \cdot b} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.7

Add $1$ and $1$ .

$30 a^{6} b + 9 a^{4} \sqrt{4 a^{2} b \cdot b} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.8

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

$30 a^{6} b + 9 a^{4} \sqrt{4 a^{2} (b^{1} b)} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.9

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

$30 a^{6} b + 9 a^{4} \sqrt{4 a^{2} (b^{1} b^{1})} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.10

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

$30 a^{6} b + 9 a^{4} \sqrt{4 a^{2} b^{1 + 1}} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.8.11

Add $1$ and $1$ .

$30 a^{6} b + 9 a^{4} \sqrt{4 a^{2} b^{2}} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.9

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

$30 a^{6} b + 9 a^{4} \sqrt{{(2 a b)}^{2}} - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.10

Pull terms out from under the radical, assuming positive real numbers.

$30 a^{6} b + 9 a^{4} (2 a b) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.11

Multiply $a^{4}$ by $a$ by adding the exponents.

Tap for more steps...

Step 2.1.11.1

Move $a$ .

$30 a^{6} b + 9 (a \cdot a^{4}) (2 b) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.11.2

Multiply $a$ by $a^{4}$ .

Tap for more steps...

Step 2.1.11.2.1

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

$30 a^{6} b + 9 (a^{1} a^{4}) (2 b) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.11.2.2

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

$30 a^{6} b + 9 a^{1 + 4} (2 b) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.11.3

Add $1$ and $4$ .

$30 a^{6} b + 9 a^{5} (2 b) - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.12

Rewrite using the commutative property of multiplication.

$30 a^{6} b + 9 \cdot 2 a^{5} b - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.13

Multiply $9$ by $2$ .

$30 a^{6} b + 18 a^{5} b - 5 a^{3} \sqrt{2 a b} (5 a^{3} \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.14

Multiply $a^{3}$ by $a^{3}$ by adding the exponents.

Tap for more steps...

Step 2.1.14.1

Move $a^{3}$ .

$30 a^{6} b + 18 a^{5} b - 5 (a^{3} a^{3}) \sqrt{2 a b} (5 \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.14.2

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

$30 a^{6} b + 18 a^{5} b - 5 a^{3 + 3} \sqrt{2 a b} (5 \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.14.3

Add $3$ and $3$ .

$30 a^{6} b + 18 a^{5} b - 5 a^{6} \sqrt{2 a b} (5 \sqrt{2 b a}) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15

Multiply $- 5 a^{6} \sqrt{2 a b} (5 \sqrt{2 b a})$ .

Tap for more steps...

Step 2.1.15.1

Multiply $5$ by $- 5$ .

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{2 a b} \sqrt{2 b a} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.2

Combine using the product rule for radicals.

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{2 b a (2 a b)} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.3

Multiply $2$ by $2$ .

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 b a (a b)} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.4

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

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 a (a (b^{1} b))} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.5

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

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 a (a (b^{1} b^{1}))} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.6

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

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 a (a b^{1 + 1})} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.7

Add $1$ and $1$ .

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 a (a b^{2})} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.8

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

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 (a^{1} a) b^{2}} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.9

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

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 (a^{1} a^{1}) b^{2}} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.10

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

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 a^{1 + 1} b^{2}} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.15.11

Add $1$ and $1$ .

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{4 a^{2} b^{2}} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.16

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

$30 a^{6} b + 18 a^{5} b - 25 a^{6} \sqrt{{(2 a b)}^{2}} - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.17

Pull terms out from under the radical, assuming positive real numbers.

$30 a^{6} b + 18 a^{5} b - 25 a^{6} (2 a b) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.18

Multiply $a^{6}$ by $a$ by adding the exponents.

Tap for more steps...

Step 2.1.18.1

Move $a$ .

$30 a^{6} b + 18 a^{5} b - 25 (a \cdot a^{6}) (2 b) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.18.2

Multiply $a$ by $a^{6}$ .

Tap for more steps...

Step 2.1.18.2.1

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

$30 a^{6} b + 18 a^{5} b - 25 (a^{1} a^{6}) (2 b) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.18.2.2

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

$30 a^{6} b + 18 a^{5} b - 25 a^{1 + 6} (2 b) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.18.3

Add $1$ and $6$ .

$30 a^{6} b + 18 a^{5} b - 25 a^{7} (2 b) - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.19

Rewrite using the commutative property of multiplication.

$30 a^{6} b + 18 a^{5} b - 25 \cdot 2 a^{7} b - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.20

Multiply $- 25$ by $2$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 5 a^{3} \sqrt{2 a b} (3 a^{2} \sqrt{2 a b})$

Step 2.1.21

Multiply $a^{3}$ by $a^{2}$ by adding the exponents.

Tap for more steps...

Step 2.1.21.1

Move $a^{2}$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 5 (a^{2} a^{3}) \sqrt{2 a b} (3 \sqrt{2 a b})$

Step 2.1.21.2

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

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 5 a^{2 + 3} \sqrt{2 a b} (3 \sqrt{2 a b})$

Step 2.1.21.3

Add $2$ and $3$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 5 a^{5} \sqrt{2 a b} (3 \sqrt{2 a b})$

Step 2.1.22

Multiply $- 5 a^{5} \sqrt{2 a b} (3 \sqrt{2 a b})$ .

Tap for more steps...

Step 2.1.22.1

Multiply $3$ by $- 5$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} \sqrt{2 a b} \sqrt{2 a b}$

Step 2.1.22.2

Raise $\sqrt{2 a b}$ to the power of $1$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} ({\sqrt{2 a b}}^{1} \sqrt{2 a b})$

Step 2.1.22.3

Raise $\sqrt{2 a b}$ to the power of $1$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} ({\sqrt{2 a b}}^{1} {\sqrt{2 a b}}^{1})$

Step 2.1.22.4

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

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} {\sqrt{2 a b}}^{1 + 1}$

Step 2.1.22.5

Add $1$ and $1$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} {\sqrt{2 a b}}^{2}$

Step 2.1.23

Rewrite ${\sqrt{2 a b}}^{2}$ as $2 a b$ .

Tap for more steps...

Step 2.1.23.1

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

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} {({(2 a b)}^{\frac{1}{2}})}^{2}$

Step 2.1.23.2

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

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} {(2 a b)}^{\frac{1}{2} \cdot 2}$

Step 2.1.23.3

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

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} {(2 a b)}^{\frac{2}{2}}$

Step 2.1.23.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.23.4.1

Cancel the common factor.

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} {(2 a b)}^{\frac{2}{2}}$

Step 2.1.23.4.2

Rewrite the expression.

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} {(2 a b)}^{1}$

Step 2.1.23.5

Simplify.

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{5} (2 a b)$

Step 2.1.24

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

Tap for more steps...

Step 2.1.24.1

Move $a$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 (a \cdot a^{5}) (2 b)$

Step 2.1.24.2

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

Tap for more steps...

Step 2.1.24.2.1

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

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 (a^{1} a^{5}) (2 b)$

Step 2.1.24.2.2

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

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{1 + 5} (2 b)$

Step 2.1.24.3

Add $1$ and $5$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 a^{6} (2 b)$

Step 2.1.25

Rewrite using the commutative property of multiplication.

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 15 \cdot 2 a^{6} b$

Step 2.1.26

Multiply $- 15$ by $2$ .

$30 a^{6} b + 18 a^{5} b - 50 a^{7} b - 30 a^{6} b$

Step 2.2

Subtract $30 a^{6} b$ from $30 a^{6} b$ .

$18 a^{5} b - 50 a^{7} b + 0$

Step 2.3

Add $18 a^{5} b$ and $0$ .

$18 a^{5} b - 50 a^{7} b$