Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $5 \sqrt{2} (5 \sqrt{2})$ .

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Step 2.1.1.1

Multiply $5$ by $5$ .

$25 \sqrt{2} \sqrt{2} + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.1.2

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

$25 ({\sqrt{2}}^{1} \sqrt{2}) + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.1.3

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

$25 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.1.4

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

$25 {\sqrt{2}}^{1 + 1} + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.1.5

Add $1$ and $1$ .

$25 {\sqrt{2}}^{2} + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.2

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

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Step 2.1.2.1

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

$25 {(2^{\frac{1}{2}})}^{2} + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.2.2

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

$25 \cdot 2^{\frac{1}{2} \cdot 2} + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.2.3

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

$25 \cdot 2^{\frac{2}{2}} + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

$25 \cdot 2^{\frac{2}{2}} + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.2.4.2

Rewrite the expression.

$25 \cdot 2^{1} + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.2.5

Evaluate the exponent.

$25 \cdot 2 + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.3

Multiply $25$ by $2$ .

$50 + 5 \sqrt{2} \cdot - 4 + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.4

Multiply $- 4$ by $5$ .

$50 - 20 \sqrt{2} + 4 (5 \sqrt{2}) + 4 \cdot - 4$

Step 2.1.5

Multiply $5$ by $4$ .

$50 - 20 \sqrt{2} + 20 \sqrt{2} + 4 \cdot - 4$

Step 2.1.6

Multiply $4$ by $- 4$ .

$50 - 20 \sqrt{2} + 20 \sqrt{2} - 16$

Step 2.2

Subtract $16$ from $50$ .

$34 - 20 \sqrt{2} + 20 \sqrt{2}$

Step 2.3

Add $- 20 \sqrt{2}$ and $20 \sqrt{2}$ .

$34 + 0$

Step 2.4

Add $34$ and $0$ .

$34$