Trigonometry Examples

$(7 \sqrt{10} + 7) (6 \sqrt{10} - 4 \sqrt{7})$

Step 1

Expand $(7 \sqrt{10} + 7) (6 \sqrt{10} - 4 \sqrt{7})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$7 \sqrt{10} (6 \sqrt{10} - 4 \sqrt{7}) + 7 (6 \sqrt{10} - 4 \sqrt{7})$

Step 1.2

Apply the distributive property.

$7 \sqrt{10} (6 \sqrt{10}) + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10} - 4 \sqrt{7})$

Step 1.3

Apply the distributive property.

$7 \sqrt{10} (6 \sqrt{10}) + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2

Simplify each term.

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Step 2.1

Multiply $7 \sqrt{10} (6 \sqrt{10})$ .

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Step 2.1.1

Multiply $6$ by $7$ .

$42 \sqrt{10} \sqrt{10} + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.1.2

Raise $\sqrt{10}$ to the power of $1$ .

$42 ({\sqrt{10}}^{1} \sqrt{10}) + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.1.3

Raise $\sqrt{10}$ to the power of $1$ .

$42 ({\sqrt{10}}^{1} {\sqrt{10}}^{1}) + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.1.4

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

$42 {\sqrt{10}}^{1 + 1} + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.1.5

Add $1$ and $1$ .

$42 {\sqrt{10}}^{2} + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$42 {(10^{\frac{1}{2}})}^{2} + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.2.2

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

$42 \cdot 10^{\frac{1}{2} \cdot 2} + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.2.3

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

$42 \cdot 10^{\frac{2}{2}} + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$42 \cdot 10^{\frac{2}{2}} + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.2.4.2

Rewrite the expression.

$42 \cdot 10^{1} + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.2.5

Evaluate the exponent.

$42 \cdot 10 + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.3

Multiply $42$ by $10$ .

$420 + 7 \sqrt{10} (- 4 \sqrt{7}) + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.4

Multiply $7 \sqrt{10} (- 4 \sqrt{7})$ .

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Step 2.4.1

Multiply $- 4$ by $7$ .

$420 - 28 \sqrt{10} \sqrt{7} + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.4.2

Combine using the product rule for radicals.

$420 - 28 \sqrt{7 \cdot 10} + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.4.3

Multiply $7$ by $10$ .

$420 - 28 \sqrt{70} + 7 (6 \sqrt{10}) + 7 (- 4 \sqrt{7})$

Step 2.5

Multiply $6$ by $7$ .

$420 - 28 \sqrt{70} + 42 \sqrt{10} + 7 (- 4 \sqrt{7})$

Step 2.6

Multiply $- 4$ by $7$ .

$420 - 28 \sqrt{70} + 42 \sqrt{10} - 28 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$420 - 28 \sqrt{70} + 42 \sqrt{10} - 28 \sqrt{7}$

Decimal Form:

$244.46981758 \dots$