Basic Math Examples

${(2 \sqrt{5} + 3 \sqrt{7})}^{2}$

Step 1

Rewrite ${(2 \sqrt{5} + 3 \sqrt{7})}^{2}$ as $(2 \sqrt{5} + 3 \sqrt{7}) (2 \sqrt{5} + 3 \sqrt{7})$ .

$(2 \sqrt{5} + 3 \sqrt{7}) (2 \sqrt{5} + 3 \sqrt{7})$

Step 2

Expand $(2 \sqrt{5} + 3 \sqrt{7}) (2 \sqrt{5} + 3 \sqrt{7})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$2 \sqrt{5} (2 \sqrt{5} + 3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5} + 3 \sqrt{7})$

Step 2.2

Apply the distributive property.

$2 \sqrt{5} (2 \sqrt{5}) + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5} + 3 \sqrt{7})$

Step 2.3

Apply the distributive property.

$2 \sqrt{5} (2 \sqrt{5}) + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

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

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Step 3.1.1.1

Multiply $2$ by $2$ .

$4 \sqrt{5} \sqrt{5} + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.1.2

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

$4 ({\sqrt{5}}^{1} \sqrt{5}) + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.1.3

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

$4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.1.4

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

$4 {\sqrt{5}}^{1 + 1} + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.1.5

Add $1$ and $1$ .

$4 {\sqrt{5}}^{2} + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.2

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

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Step 3.1.2.1

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

$4 {(5^{\frac{1}{2}})}^{2} + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.2.2

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

$4 \cdot 5^{\frac{1}{2} \cdot 2} + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.2.3

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

$4 \cdot 5^{\frac{2}{2}} + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

$4 \cdot 5^{\frac{2}{2}} + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.2.4.2

Rewrite the expression.

$4 \cdot 5^{1} + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.2.5

Evaluate the exponent.

$4 \cdot 5 + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.3

Multiply $4$ by $5$ .

$20 + 2 \sqrt{5} (3 \sqrt{7}) + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.4

Multiply $2 \sqrt{5} (3 \sqrt{7})$ .

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Step 3.1.4.1

Multiply $3$ by $2$ .

$20 + 6 \sqrt{5} \sqrt{7} + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.4.2

Combine using the product rule for radicals.

$20 + 6 \sqrt{7 \cdot 5} + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.4.3

Multiply $7$ by $5$ .

$20 + 6 \sqrt{35} + 3 \sqrt{7} (2 \sqrt{5}) + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.5

Multiply $3 \sqrt{7} (2 \sqrt{5})$ .

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Step 3.1.5.1

Multiply $2$ by $3$ .

$20 + 6 \sqrt{35} + 6 \sqrt{7} \sqrt{5} + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.5.2

Combine using the product rule for radicals.

$20 + 6 \sqrt{35} + 6 \sqrt{5 \cdot 7} + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.5.3

Multiply $5$ by $7$ .

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 3 \sqrt{7} (3 \sqrt{7})$

Step 3.1.6

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

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Step 3.1.6.1

Multiply $3$ by $3$ .

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 \sqrt{7} \sqrt{7}$

Step 3.1.6.2

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

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 ({\sqrt{7}}^{1} \sqrt{7})$

Step 3.1.6.3

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

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})$

Step 3.1.6.4

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

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 {\sqrt{7}}^{1 + 1}$

Step 3.1.6.5

Add $1$ and $1$ .

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 {\sqrt{7}}^{2}$

Step 3.1.7

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

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Step 3.1.7.1

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

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 {(7^{\frac{1}{2}})}^{2}$

Step 3.1.7.2

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

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 \cdot 7^{\frac{1}{2} \cdot 2}$

Step 3.1.7.3

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

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 \cdot 7^{\frac{2}{2}}$

Step 3.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.7.4.1

Cancel the common factor.

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 \cdot 7^{\frac{2}{2}}$

Step 3.1.7.4.2

Rewrite the expression.

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 \cdot 7^{1}$

Step 3.1.7.5

Evaluate the exponent.

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 9 \cdot 7$

Step 3.1.8

Multiply $9$ by $7$ .

$20 + 6 \sqrt{35} + 6 \sqrt{35} + 63$

Step 3.2

Add $20$ and $63$ .

$83 + 6 \sqrt{35} + 6 \sqrt{35}$

Step 3.3

Add $6 \sqrt{35}$ and $6 \sqrt{35}$ .

$83 + 12 \sqrt{35}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$83 + 12 \sqrt{35}$

Decimal Form:

$153.99295739 \dots$