Algebra Examples

$\sqrt{4 + \sqrt{7}} \cdot \sqrt{4 - \sqrt{7}}$

Step 1

Combine using the product rule for radicals.

$\sqrt{(4 + \sqrt{7}) \cdot (4 - \sqrt{7})}$

Step 2

Expand $(4 + \sqrt{7}) (4 - \sqrt{7})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{4 (4 - \sqrt{7}) + \sqrt{7} (4 - \sqrt{7})}$

Step 2.2

Apply the distributive property.

$\sqrt{4 \cdot 4 + 4 (- \sqrt{7}) + \sqrt{7} (4 - \sqrt{7})}$

Step 2.3

Apply the distributive property.

$\sqrt{4 \cdot 4 + 4 (- \sqrt{7}) + \sqrt{7} \cdot 4 + \sqrt{7} (- \sqrt{7})}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $4$ by $4$ .

$\sqrt{16 + 4 (- \sqrt{7}) + \sqrt{7} \cdot 4 + \sqrt{7} (- \sqrt{7})}$

Step 3.1.2

Multiply $- 1$ by $4$ .

$\sqrt{16 - 4 \sqrt{7} + \sqrt{7} \cdot 4 + \sqrt{7} (- \sqrt{7})}$

Step 3.1.3

Move $4$ to the left of $\sqrt{7}$ .

$\sqrt{16 - 4 \sqrt{7} + 4 \cdot \sqrt{7} + \sqrt{7} (- \sqrt{7})}$

Step 3.1.4

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

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Step 3.1.4.1

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - ({\sqrt{7}}^{1} \sqrt{7})}$

Step 3.1.4.2

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}$

Step 3.1.4.3

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - {\sqrt{7}}^{1 + 1}}$

Step 3.1.4.4

Add $1$ and $1$ .

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - {\sqrt{7}}^{2}}$

Step 3.1.5

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

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Step 3.1.5.1

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - {(7^{\frac{1}{2}})}^{2}}$

Step 3.1.5.2

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7^{\frac{1}{2} \cdot 2}}$

Step 3.1.5.3

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7^{\frac{2}{2}}}$

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7^{\frac{2}{2}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7^{1}}$

Step 3.1.5.5

Evaluate the exponent.

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 1 \cdot 7}$

Step 3.1.6

Multiply $- 1$ by $7$ .

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7}$

Step 3.2

Subtract $7$ from $16$ .

$\sqrt{9 - 4 \sqrt{7} + 4 \sqrt{7}}$

Step 3.3

Add $- 4 \sqrt{7}$ and $4 \sqrt{7}$ .

$\sqrt{9 + 0}$

Step 3.4

Add $9$ and $0$ .

$\sqrt{9}$

Step 4

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Step 5

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

$3$