Simplify Binomial Radical Expressions using the FOIL Method 1

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Question 1 of 3

1. Question

Simplify

$(\sqrt{2} + 3) (\sqrt{2} - 4)$ $(sqrt2+3) (sqrt2-4)$

1. $- \sqrt{2} - 10$ $- sqrt2 - 10$
2. $- 10$ $- 10$
3. $- 10 \sqrt{2}$ $-10sqrt2$
4. $\sqrt{2} + 10$ $sqrt2 + 10$

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Multiplication Property of Radicals

\sqrt{a b} = \sqrt{a} \times \sqrt{b}

$sqrt(ab)=sqrt(a) xx sqrt(b)$

Distributive Property

(a + b) (c + d) = a c + a d + b c + b d

$(color(darkviolet)(a)+color(royalblue)(b)) (color(forestgreen)(c)+color(crimson)(d))=color(darkviolet)(a)color(forestgreen)(c)+color(darkviolet)(a)color(crimson)(d)+color(royalblue)(b)color(forestgreen)(c)+color(royalblue)(b)color(crimson)(d)$

Apply the Distributive Property.

$=$ $=$	$(\sqrt{2} + 3) (\sqrt{2} - 4)$ $(color(darkviolet)(sqrt2)+color(royalblue)(3)) (color(forestgreen)(sqrt2)-color(crimson)(4))$
$=$ $=$	$\sqrt{2} \times \sqrt{2} - \sqrt{2} \times 4 + 3 \times \sqrt{2} - 34$ $color(darkviolet)(sqrt2) xx color(forestgreen)(sqrt2)-color(darkviolet)(sqrt2) xx color(crimson)(4)+color(royalblue)(3) xx color(forestgreen)(sqrt2)-color(royalblue)(3)color(crimson)(4)$
$=$ $=$	$\sqrt{4} - 4 \sqrt{2} + 3 \sqrt{2} - 12$ $sqrt4 – 4sqrt2 + 3sqrt2 – 12$	Simplify like terms (same radicand)
$=$ $=$	$2 - \sqrt{2} - 12$ $2 – sqrt2 – 12$	Simplify like terms (constants)
$=$ $=$	$- \sqrt{2} - 10$ $- sqrt2 – 10$

- \sqrt{2} - 10

$- sqrt2 – 10$

Question 2 of 3

2. Question

Simplify

$(3 \sqrt{2} - \sqrt{6}) (2 \sqrt{2} + 3 \sqrt{6})$ $(3sqrt2-sqrt6) (2sqrt2+3sqrt6)$

1. $14 + \sqrt{6}$ $14+sqrt6$
2. $14 \sqrt{3} - 6$ $14sqrt3- 6$
3. $\sqrt{3} - 6$ $sqrt3- 6$
4. $8 \sqrt{3}$ $8sqrt3$

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Multiplication Property of Radicals

\sqrt{a b} = \sqrt{a} \times \sqrt{b}

$sqrt(ab)=sqrt(a) xx sqrt(b)$

Distributive Property

(a + b) (c + d) = a c + a d + b c + b d

Apply the Distributive Property.

$=$ $=$	$(3 \sqrt{2} - \sqrt{6}) (2 \sqrt{2} + 3 \sqrt{6})$ $(color(darkviolet)(3sqrt2)-color(royalblue)(sqrt6)) (color(forestgreen)(2sqrt2)+color(crimson)(3sqrt6))$
$=$ $=$	$3 \sqrt{2} \times 2 \sqrt{2} + 3 \sqrt{2} \times 3 \sqrt{6} - \sqrt{6} \times \sqrt{2} - \sqrt{6} \times 3 \sqrt{6}$ $color(darkviolet)(3sqrt2) xx color(forestgreen)(2sqrt2)+color(darkviolet)(3sqrt2) xx color(crimson)(3sqrt6)-color(royalblue)(sqrt6) xx color(forestgreen)(sqrt2)-color(royalblue)(sqrt6) xx color(crimson)(3sqrt6)$
$=$ $=$	$6 \sqrt{4} + 9 \sqrt{12} - 2 \sqrt{12} - 3 \sqrt{36}$ $6sqrt4 + 9sqrt12 -2sqrt12-3sqrt36$	Simplify the Radicals
$=$ $=$	$6 \times 2 + 9 \times 2 \sqrt{3} - 2 \times 2 \sqrt{3} - 3 \times 6$ $6 xx 2 + 9 xx 2sqrt3 – 2 xx 2sqrt3 – 3 xx 6$	Multiplying
$=$ $=$	$12 + 18 \sqrt{3} - 4 \sqrt{3} - 18$ $12 + 18sqrt3-4sqrt3-18$	Combine like Terms
$=$ $=$	$14 \sqrt{3} - 6$ $14sqrt3- 6$

14 \sqrt{3} - 6

$14sqrt3- 6$

Question 3 of 3

3. Question

Simplify

$(\sqrt{7} - 5 \sqrt{2}) (\sqrt{2} - 5 \sqrt{7})$ $(sqrt7-5sqrt2) (sqrt2-5sqrt7)$

1. $26 \sqrt{14} - 45$ $26sqrt14- 45$
2. $26 \sqrt{14} + 45$ $26sqrt14+ 45$
3. $45 \sqrt{14}$ $45sqrt14$
4. $45 - 26 \sqrt{14}$ $45-26sqrt14$

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Multiplication Property of Radicals

\sqrt{a b} = \sqrt{a} \times \sqrt{b}

$sqrt(ab)=sqrt(a) xx sqrt(b)$

Distributive Property

(a + b) (c + d) = a c + a d + b c + b d

Apply the Distributive Property.

$=$	$(color(darkviolet)(sqrt7)-color(royalblue)(5sqrt2)) (color(forestgreen)(sqrt2)-color(crimson)(5sqrt7))$
$=$	$color(darkviolet)(sqrt7) xx color(forestgreen)(sqrt2)-color(darkviolet)(sqrt7) xx color(crimson)(5sqrt7)-color(royalblue)(5sqrt2) xx color(forestgreen)(sqrt2)+color(royalblue)(5sqrt2) xx color(crimson)(5sqrt7)$
$=$	$sqrt14 – 5sqrt49 – 5sqrt4 + 25sqrt14$	Simplify the Radicals
$=$	$sqrt14 – 5 xx 7- 5 xx 2 + 25sqrt14$	Multiplying
$=$	$sqrt14 – 35- 10 + 25sqrt14$	Combine like Terms
$=$	$26sqrt14- 45$

$26sqrt14- 45$

Quizzes

Simplify Square Roots 1
Simplify Square Roots 2
Simplify Square Roots 3
Simplify Square Roots 4
Simplify Radicals with Variables 1
Simplify Radicals with Variables 2
Simplify Radicals with Variables 3
Rewriting Entire and Mixed Radicals 1
Rewriting Entire and Mixed Radicals 2
Add and Subtract Radical Expressions (Basic) 1
Add and Subtract Radical Expressions (Basic) 2
Add and Subtract Radical Expressions (Basic) 3
Add and Subtract Radical Expressions 1
Add and Subtract Radical Expressions 2
Add and Subtract Radical Expressions 3
Multiply Radical Expressions 1
Multiply Radical Expressions 2
Multiply Radical Expressions 3
Multiply Radical Expressions 4
Divide Radical Expressions 1
Divide Radical Expressions 2
Divide Radical Expressions 3
Multiply and Divide Radical Expressions
Simplify Radical Expressions using the Distributive Property 1
Simplify Radical Expressions using the Distributive Property 2
Simplify Radical Expressions using the Distributive Property 3
Simplify Binomial Radical Expressions using the FOIL Method 1
Simplify Binomial Radical Expressions using the FOIL Method 2
Rationalizing the Denominator 1
Rationalizing the Denominator 2
Rationalizing the Denominator 3
Rationalizing the Denominator 4
Rationalizing the Denominator using Conjugates