Simplify the Radical Expression (10+3 square root of 7)(10-3 square root of 7)

$(10 + 3 \sqrt{7}) (10 - 3 \sqrt{7})$

Step 1

Expand $(10 + 3 \sqrt{7}) (10 - 3 \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$10 (10 - 3 \sqrt{7}) + 3 \sqrt{7} (10 - 3 \sqrt{7})$

Apply the distributive property.

$10 \cdot 10 + 10 (- 3 \sqrt{7}) + 3 \sqrt{7} (10 - 3 \sqrt{7})$

Apply the distributive property.

$10 \cdot 10 + 10 (- 3 \sqrt{7}) + 3 \sqrt{7} \cdot 10 + 3 \sqrt{7} (- 3 \sqrt{7})$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $10$ by $10$ .

$100 + 10 (- 3 \sqrt{7}) + 3 \sqrt{7} \cdot 10 + 3 \sqrt{7} (- 3 \sqrt{7})$

Multiply $- 3$ by $10$ .

$100 - 30 \sqrt{7} + 3 \sqrt{7} \cdot 10 + 3 \sqrt{7} (- 3 \sqrt{7})$

Multiply $10$ by $3$ .

$100 - 30 \sqrt{7} + 30 \sqrt{7} + 3 \sqrt{7} (- 3 \sqrt{7})$

Multiply $3 \sqrt{7} (- 3 \sqrt{7})$ .

Multiply $- 3$ by $3$ .

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 \sqrt{7} \sqrt{7}$

Raise $\sqrt{7}$ to the power of $1$ .

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 ({\sqrt{7}}^{1} \sqrt{7})$

Raise $\sqrt{7}$ to the power of $1$ .

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})$

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

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 {\sqrt{7}}^{1 + 1}$

Add $1$ and $1$ .

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 {\sqrt{7}}^{2}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 {(7^{\frac{1}{2}})}^{2}$

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

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 \cdot 7^{\frac{1}{2} \cdot 2}$

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

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 \cdot 7^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 \cdot 7^{\frac{2}{2}}$

Rewrite the expression.

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 \cdot 7^{1}$

Evaluate the exponent.

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 9 \cdot 7$

Multiply $- 9$ by $7$ .

$100 - 30 \sqrt{7} + 30 \sqrt{7} - 63$

Subtract $63$ from $100$ .

$37 - 30 \sqrt{7} + 30 \sqrt{7}$

Add $- 30 \sqrt{7}$ and $30 \sqrt{7}$ .

$37 + 0$

Add $37$ and $0$ .

$37$

Simplify the Radical Expression (10+3 square root of 7)(10-3 square root of 7)

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