Algebra Examples

$(5 \sqrt{2 x} + \sqrt{5}) (- 4 \sqrt{2 x} + \sqrt{5 x})$

Step 1

Expand $(5 \sqrt{2 x} + \sqrt{5}) (- 4 \sqrt{2 x} + \sqrt{5 x})$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

$5 \sqrt{2 x} (- 4 \sqrt{2 x} + \sqrt{5 x}) + \sqrt{5} (- 4 \sqrt{2 x} + \sqrt{5 x})$

Step 1.2

Apply the distributive property.

$5 \sqrt{2 x} (- 4 \sqrt{2 x}) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x} + \sqrt{5 x})$

Step 1.3

Apply the distributive property.

$5 \sqrt{2 x} (- 4 \sqrt{2 x}) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Multiply $5 \sqrt{2 x} (- 4 \sqrt{2 x})$ .

Tap for more steps...

Step 2.1.1

Multiply $- 4$ by $5$ .

$- 20 \sqrt{2 x} \sqrt{2 x} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.1.2

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

$- 20 ({\sqrt{2 x}}^{1} \sqrt{2 x}) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.1.3

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

$- 20 ({\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.1.4

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

$- 20 {\sqrt{2 x}}^{1 + 1} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.1.5

Add $1$ and $1$ .

$- 20 {\sqrt{2 x}}^{2} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.2

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

Tap for more steps...

Step 2.2.1

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

$- 20 {({(2 x)}^{\frac{1}{2}})}^{2} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.2.2

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

$- 20 {(2 x)}^{\frac{1}{2} \cdot 2} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.2.3

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

$- 20 {(2 x)}^{\frac{2}{2}} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.4.1

Cancel the common factor.

$- 20 {(2 x)}^{\frac{2}{2}} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.2.4.2

Rewrite the expression.

$- 20 {(2 x)}^{1} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.2.5

Simplify.

$- 20 (2 x) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.3

Multiply $2$ by $- 20$ .

$- 40 x + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.4

Multiply $5 \sqrt{2 x} \sqrt{5 x}$ .

Tap for more steps...

Step 2.4.1

Combine using the product rule for radicals.

$- 40 x + 5 \sqrt{5 x (2 x)} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.4.2

Multiply $2$ by $5$ .

$- 40 x + 5 \sqrt{10 x \cdot x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.4.3

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

$- 40 x + 5 \sqrt{10 (x^{1} x)} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.4.4

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

$- 40 x + 5 \sqrt{10 (x^{1} x^{1})} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.4.5

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

$- 40 x + 5 \sqrt{10 x^{1 + 1}} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.4.6

Add $1$ and $1$ .

$- 40 x + 5 \sqrt{10 x^{2}} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.5

Reorder $10$ and $x^{2}$ .

$- 40 x + 5 \sqrt{x^{2} \cdot 10} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.6

Pull terms out from under the radical.

$- 40 x + 5 (x \sqrt{10}) + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2.7

Multiply $\sqrt{5} (- 4 \sqrt{2 x})$ .

Tap for more steps...

Step 2.7.1

Combine using the product rule for radicals.

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{5 (2 x)} + \sqrt{5} \sqrt{5 x}$

Step 2.7.2

Multiply $5$ by $2$ .

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + \sqrt{5} \sqrt{5 x}$

Step 2.8

Multiply $\sqrt{5} \sqrt{5 x}$ .

Tap for more steps...

Step 2.8.1

Combine using the product rule for radicals.

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + \sqrt{5 (5 x)}$

Step 2.8.2

Multiply $5$ by $5$ .

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + \sqrt{25 x}$

Step 2.9

Rewrite $25$ as $5^{2}$ .

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + \sqrt{5^{2} x}$

Step 2.10

Pull terms out from under the radical.

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + 5 \sqrt{x}$