Algebra Examples

${(4 x \sqrt{5 x^{2}} + 2 x^{2} \sqrt{6})}^{2}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Reorder $5$ and $x^{2}$ .

${(4 x \sqrt{x^{2} \cdot 5} + 2 x^{2} \sqrt{6})}^{2}$

Step 1.1.2

Pull terms out from under the radical.

${(4 x (x \sqrt{5}) + 2 x^{2} \sqrt{6})}^{2}$

Step 1.1.3

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

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Step 1.1.3.1

Move $x$ .

${(4 (x \cdot x) \sqrt{5} + 2 x^{2} \sqrt{6})}^{2}$

Step 1.1.3.2

Multiply $x$ by $x$ .

${(4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6})}^{2}$

Step 1.2

Rewrite ${(4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6})}^{2}$ as $(4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6}) (4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6})$ .

$(4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6}) (4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6})$

Step 2

Expand $(4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6}) (4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$4 x^{2} \sqrt{5} (4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6})$

Step 2.2

Apply the distributive property.

$4 x^{2} \sqrt{5} (4 x^{2} \sqrt{5}) + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5} + 2 x^{2} \sqrt{6})$

Step 2.3

Apply the distributive property.

$4 x^{2} \sqrt{5} (4 x^{2} \sqrt{5}) + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

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 $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.1.1.1

Move $x^{2}$ .

$4 (x^{2} x^{2}) \sqrt{5} (4 \sqrt{5}) + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.1.2

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

$4 x^{2 + 2} \sqrt{5} (4 \sqrt{5}) + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.1.3

Add $2$ and $2$ .

$4 x^{4} \sqrt{5} (4 \sqrt{5}) + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.2

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

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Step 3.1.2.1

Multiply $4$ by $4$ .

$16 x^{4} \sqrt{5} \sqrt{5} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$16 x^{4} ({\sqrt{5}}^{1} \sqrt{5}) + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.2.3

Raise $\sqrt{5}$ to the power of $1$ .

$16 x^{4} ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.2.4

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

$16 x^{4} {\sqrt{5}}^{1 + 1} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.2.5

Add $1$ and $1$ .

$16 x^{4} {\sqrt{5}}^{2} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.3

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$16 x^{4} {(5^{\frac{1}{2}})}^{2} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.3.2

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

$16 x^{4} \cdot 5^{\frac{1}{2} \cdot 2} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.3.3

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

$16 x^{4} \cdot 5^{\frac{2}{2}} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

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.

$16 x^{4} \cdot 5^{\frac{2}{2}} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.3.4.2

Rewrite the expression.

$16 x^{4} \cdot 5^{1} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.3.5

Evaluate the exponent.

$16 x^{4} \cdot 5 + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.4

Multiply $5$ by $16$ .

$80 x^{4} + 4 x^{2} \sqrt{5} (2 x^{2} \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.1.5.1

Move $x^{2}$ .

$80 x^{4} + 4 (x^{2} x^{2}) \sqrt{5} (2 \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.5.2

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

$80 x^{4} + 4 x^{2 + 2} \sqrt{5} (2 \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.5.3

Add $2$ and $2$ .

$80 x^{4} + 4 x^{4} \sqrt{5} (2 \sqrt{6}) + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.6

Multiply $4 x^{4} \sqrt{5} (2 \sqrt{6})$ .

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Step 3.1.6.1

Multiply $2$ by $4$ .

$80 x^{4} + 8 x^{4} \sqrt{5} \sqrt{6} + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.6.2

Combine using the product rule for radicals.

$80 x^{4} + 8 x^{4} \sqrt{6 \cdot 5} + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.6.3

Multiply $6$ by $5$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 2 x^{2} \sqrt{6} (4 x^{2} \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.7

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.1.7.1

Move $x^{2}$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 2 (x^{2} x^{2}) \sqrt{6} (4 \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.7.2

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

$80 x^{4} + 8 x^{4} \sqrt{30} + 2 x^{2 + 2} \sqrt{6} (4 \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.7.3

Add $2$ and $2$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 2 x^{4} \sqrt{6} (4 \sqrt{5}) + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.8

Multiply $2 x^{4} \sqrt{6} (4 \sqrt{5})$ .

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Step 3.1.8.1

Multiply $4$ by $2$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{6} \sqrt{5} + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.8.2

Combine using the product rule for radicals.

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{5 \cdot 6} + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.8.3

Multiply $5$ by $6$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 2 x^{2} \sqrt{6} (2 x^{2} \sqrt{6})$

Step 3.1.9

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.1.9.1

Move $x^{2}$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 2 (x^{2} x^{2}) \sqrt{6} (2 \sqrt{6})$

Step 3.1.9.2

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

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 2 x^{2 + 2} \sqrt{6} (2 \sqrt{6})$

Step 3.1.9.3

Add $2$ and $2$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 2 x^{4} \sqrt{6} (2 \sqrt{6})$

Step 3.1.10

Multiply $2 x^{4} \sqrt{6} (2 \sqrt{6})$ .

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Step 3.1.10.1

Multiply $2$ by $2$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} \sqrt{6} \sqrt{6}$

Step 3.1.10.2

Raise $\sqrt{6}$ to the power of $1$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} ({\sqrt{6}}^{1} \sqrt{6})$

Step 3.1.10.3

Raise $\sqrt{6}$ to the power of $1$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} ({\sqrt{6}}^{1} {\sqrt{6}}^{1})$

Step 3.1.10.4

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

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} {\sqrt{6}}^{1 + 1}$

Step 3.1.10.5

Add $1$ and $1$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} {\sqrt{6}}^{2}$

Step 3.1.11

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} {(6^{\frac{1}{2}})}^{2}$

Step 3.1.11.2

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

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} \cdot 6^{\frac{1}{2} \cdot 2}$

Step 3.1.11.3

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

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} \cdot 6^{\frac{2}{2}}$

Step 3.1.11.4

Cancel the common factor of $2$ .

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Step 3.1.11.4.1

Cancel the common factor.

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} \cdot 6^{\frac{2}{2}}$

Step 3.1.11.4.2

Rewrite the expression.

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} \cdot 6^{1}$

Step 3.1.11.5

Evaluate the exponent.

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 4 x^{4} \cdot 6$

Step 3.1.12

Multiply $6$ by $4$ .

$80 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30} + 24 x^{4}$

Step 3.2

Add $80 x^{4}$ and $24 x^{4}$ .

$104 x^{4} + 8 x^{4} \sqrt{30} + 8 x^{4} \sqrt{30}$

Step 3.3

Add $8 x^{4} \sqrt{30}$ and $8 x^{4} \sqrt{30}$ .

$104 x^{4} + 16 x^{4} \sqrt{30}$