Algebra Examples

$(x \sqrt{7} - 3 \sqrt{8}) (x \sqrt{7} - 3 \sqrt{8})$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1.1

Factor $4$ out of $8$ .

$(x \sqrt{7} - 3 \sqrt{4 (2)}) (x \sqrt{7} - 3 \sqrt{8})$

Step 1.1.1.2

Rewrite $4$ as $2^{2}$ .

$(x \sqrt{7} - 3 \sqrt{2^{2} \cdot 2}) (x \sqrt{7} - 3 \sqrt{8})$

Step 1.1.2

Pull terms out from under the radical.

$(x \sqrt{7} - 3 (2 \sqrt{2})) (x \sqrt{7} - 3 \sqrt{8})$

Step 1.1.3

Multiply $2$ by $- 3$ .

$(x \sqrt{7} - 6 \sqrt{2}) (x \sqrt{7} - 3 \sqrt{8})$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.1.1

Factor $4$ out of $8$ .

$(x \sqrt{7} - 6 \sqrt{2}) (x \sqrt{7} - 3 \sqrt{4 (2)})$

Step 1.2.1.2

Rewrite $4$ as $2^{2}$ .

$(x \sqrt{7} - 6 \sqrt{2}) (x \sqrt{7} - 3 \sqrt{2^{2} \cdot 2})$

Step 1.2.2

Pull terms out from under the radical.

$(x \sqrt{7} - 6 \sqrt{2}) (x \sqrt{7} - 3 (2 \sqrt{2}))$

Step 1.2.3

Multiply $2$ by $- 3$ .

$(x \sqrt{7} - 6 \sqrt{2}) (x \sqrt{7} - 6 \sqrt{2})$

Step 2

Expand $(x \sqrt{7} - 6 \sqrt{2}) (x \sqrt{7} - 6 \sqrt{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x \sqrt{7} (x \sqrt{7} - 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7} - 6 \sqrt{2})$

Step 2.2

Apply the distributive property.

$x \sqrt{7} (x \sqrt{7}) + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7} - 6 \sqrt{2})$

Step 2.3

Apply the distributive property.

$x \sqrt{7} (x \sqrt{7}) + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

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 $x$ by $x$ by adding the exponents.

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Step 3.1.1.1

Move $x$ .

$x \cdot x \sqrt{7} \sqrt{7} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.1.2

Multiply $x$ by $x$ .

$x^{2} \sqrt{7} \sqrt{7} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.2

Multiply $x^{2} \sqrt{7} \sqrt{7}$ .

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Step 3.1.2.1

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

$x^{2} ({\sqrt{7}}^{1} \sqrt{7}) + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.2.2

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

$x^{2} ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.2.3

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

$x^{2} {\sqrt{7}}^{1 + 1} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.2.4

Add $1$ and $1$ .

$x^{2} {\sqrt{7}}^{2} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.3

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

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Step 3.1.3.1

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

$x^{2} {(7^{\frac{1}{2}})}^{2} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.3.2

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

$x^{2} \cdot 7^{\frac{1}{2} \cdot 2} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.3.3

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

$x^{2} \cdot 7^{\frac{2}{2}} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.3.4.1

Cancel the common factor.

$x^{2} \cdot 7^{\frac{2}{2}} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.3.4.2

Rewrite the expression.

$x^{2} \cdot 7^{1} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.3.5

Evaluate the exponent.

$x^{2} \cdot 7 + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.4

Move $7$ to the left of $x^{2}$ .

$7 \cdot x^{2} + x \sqrt{7} (- 6 \sqrt{2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.5

Multiply $x \sqrt{7} (- 6 \sqrt{2})$ .

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Step 3.1.5.1

Combine using the product rule for radicals.

$7 x^{2} + x (- 6 \sqrt{7 \cdot 2}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.5.2

Multiply $7$ by $2$ .

$7 x^{2} + x (- 6 \sqrt{14}) - 6 \sqrt{2} (x \sqrt{7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.6

Multiply $- 6 \sqrt{2} (x \sqrt{7})$ .

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Step 3.1.6.1

Combine using the product rule for radicals.

$7 x^{2} + x (- 6 \sqrt{14}) - 6 (x \sqrt{2 \cdot 7}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.6.2

Multiply $2$ by $7$ .

$7 x^{2} + x (- 6 \sqrt{14}) - 6 (x \sqrt{14}) - 6 \sqrt{2} (- 6 \sqrt{2})$

Step 3.1.7

Multiply $- 6 \sqrt{2} (- 6 \sqrt{2})$ .

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Step 3.1.7.1

Multiply $- 6$ by $- 6$ .

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 \sqrt{2} \sqrt{2}$

Step 3.1.7.2

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

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 ({\sqrt{2}}^{1} \sqrt{2})$

Step 3.1.7.3

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

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 3.1.7.4

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

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 {\sqrt{2}}^{1 + 1}$

Step 3.1.7.5

Add $1$ and $1$ .

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 {\sqrt{2}}^{2}$

Step 3.1.8

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

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Step 3.1.8.1

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

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 {(2^{\frac{1}{2}})}^{2}$

Step 3.1.8.2

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

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 3.1.8.3

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

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 \cdot 2^{\frac{2}{2}}$

Step 3.1.8.4

Cancel the common factor of $2$ .

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Step 3.1.8.4.1

Cancel the common factor.

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 \cdot 2^{\frac{2}{2}}$

Step 3.1.8.4.2

Rewrite the expression.

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 \cdot 2^{1}$

Step 3.1.8.5

Evaluate the exponent.

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 36 \cdot 2$

Step 3.1.9

Multiply $36$ by $2$ .

$7 x^{2} + x (- 6 \sqrt{14}) - 6 x \sqrt{14} + 72$

Step 3.2

Subtract $6 x \sqrt{14}$ from $x (- 6 \sqrt{14})$ .

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Step 3.2.1

Reorder $x$ and $- 6$ .

$7 x^{2} - 6 \cdot x \sqrt{14} - 6 x \sqrt{14} + 72$

Step 3.2.2

Subtract $6 x \sqrt{14}$ from $- 6 \cdot x \sqrt{14}$ .

$7 x^{2} - 12 x \sqrt{14} + 72$