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Pearson Algebra 1 Common Core, 2011

PA

Pearson Algebra 1 Common Core, 2011 View details

3. Operations With Radical Expressions

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Exercise 43 Page 630

The Multiplication Property of Square Roots tells us that sqrt(ab)=sqrt(a) * sqrt(b), for a≥ 0 and b≥ 0.

Exact Solution: 9+6sqrt(2)+3sqrt(10)+4sqrt(5)
Approximated Solution: 35.9

Practice makes perfect

To solve the given proportion involving radicals, we will use the Multiplication Property of Square Roots. sqrt(ab)=sqrt(a) * sqrt(b), for a≥ 0,b≥ 0 Let's start by isolating the variable x.

2+sqrt(2)/2-sqrt(2)=x/3+sqrt(10)

LHS * (3+sqrt(10))=RHS* (3+sqrt(10))

2+sqrt(2)/2-sqrt(2)(3+sqrt(10))=x

MoveRightFacToNum

a/c* b = a* b/c

(2+sqrt(2))(3+sqrt(10))/2-sqrt(2)=x

▼

Simplify left-hand side

Distribute (3+sqrt(10))

2(3+sqrt(10))+sqrt(2)(3+sqrt(10))/2-sqrt(2)=x

Distribute 2 & sqrt(2)

6+2sqrt(10)+3sqrt(2)+sqrt(2)sqrt(10)/2-sqrt(2)=x

sqrt(a)*sqrt(b)=sqrt(a* b)

6+2sqrt(10)+3sqrt(2)+sqrt(20)/2-sqrt(2)=x

SplitIntoFactors

Split into factors

6+2sqrt(10)+3sqrt(2)+sqrt(4 * 5)/2-sqrt(2)=x

sqrt(a* b)=sqrt(a)*sqrt(b)

6+2sqrt(10)+3sqrt(2)+sqrt(4)sqrt(5)/2-sqrt(2)=x

Calculate root

6+2sqrt(10)+3sqrt(2)+2sqrt(5)/2-sqrt(2)=x

Rearrange equation

x=6+2sqrt(10)+3sqrt(2)+2sqrt(5)/2-sqrt(2)

Next, we need to rationalize the denominator of the right-hand side of the above equation. This is done by multiplying the numerator and the denominator by the conjugate of the denominator. We find the conjugate by changing the sign of the irrational part. Denominator:& 2 - sqrt(2) Conjugate:& 2 + sqrt(2) Notice that multiplying both the numerator and the denominator of the expression will not change its value, because anything divided by itself is 1. x=6+2sqrt(10)+3sqrt(2)+2sqrt(5)/2-sqrt(2) ⇕ x=6+2sqrt(10)+3sqrt(2)+2sqrt(5)/2-sqrt(2)* 2+sqrt(2)/2+sqrt(2) We can simplify the right-hand side to find the value of x. Let's first simplify the numerator.

x=6+2sqrt(10)+3sqrt(2)+2sqrt(5)/2-sqrt(2)* 2+sqrt(2)/2+sqrt(2)

Multiply fractions

x=(6+2sqrt(10)+3sqrt(2)+2sqrt(5))(2+sqrt(2))/(2-sqrt(2)) (2+sqrt(2))

▼

Multiply

Distribute (2+sqrt(2))

x=6(2+sqrt(2))+2sqrt(10)(2+sqrt(2))+3sqrt(2)(2+sqrt(2))+2sqrt(5)(2+sqrt(2))/(2-sqrt(2)) (2+sqrt(2))

Distribute 6 & 2sqrt(10) & 3sqrt(2) & 2sqrt(5)

x=12+6sqrt(2)+4sqrt(10)+2sqrt(10)sqrt(2)+6sqrt(2)+3sqrt(2)sqrt(2)+4sqrt(5)+2sqrt(5)sqrt(2)/(2-sqrt(2)) (2+sqrt(2))

▼

Simplify

sqrt(a)*sqrt(b)=sqrt(a* b)

x=12+6sqrt(2)+4sqrt(10)+2sqrt(20)+6sqrt(2)+3sqrt(4)+4sqrt(5)+2sqrt(10)/(2-sqrt(2)) (2+sqrt(2))

SplitIntoFactors

Split into factors

x=12+6sqrt(2)+4sqrt(10)+2sqrt(4 * 5)+6sqrt(2)+3sqrt(4)+4sqrt(5)+2sqrt(10)/(2-sqrt(2)) (2+sqrt(2))

sqrt(a* b)=sqrt(a)*sqrt(b)

x=12+6sqrt(2)+4sqrt(10)+2sqrt(4)sqrt(5)+6sqrt(2)+3sqrt(4)+4sqrt(5)+2sqrt(10)/(2-sqrt(2)) (2+sqrt(2))

Calculate root

x=12+6sqrt(2)+4sqrt(10)+2(2)sqrt(5)+6sqrt(2)+3(2)+4sqrt(5)+2sqrt(10)/(2-sqrt(2)) (2+sqrt(2))

Multiply

x=12+6sqrt(2)+4sqrt(10)+4sqrt(5)+6sqrt(2)+6+4sqrt(5)+2sqrt(10)/(2-sqrt(2)) (2+sqrt(2))

CommutativePropAdd

Commutative Property of Addition

x=12+6+6sqrt(2)+6sqrt(2)+4sqrt(10)+2sqrt(10)+4sqrt(5)+4sqrt(5)/(2-sqrt(2)) (2+sqrt(2))

Add terms

x=18+12sqrt(2)+6sqrt(10)+8sqrt(5)/(2-sqrt(2)) (2+sqrt(2))

Now that we have simplified the numerator as much as possible, let's continue with the denominator.

x=18+12sqrt(2)+6sqrt(10)+8sqrt(5)/(2-sqrt(2)) (2+sqrt(2))

(a-b)(a+b)=a^2-b^2

x=18+12sqrt(2)+6sqrt(10)+8sqrt(5)/2^2-(sqrt(2))^2

▼

Simplify

Calculate power

x=18+12sqrt(2)+6sqrt(10)+8sqrt(5)/4-(sqrt(2))^2

( sqrt(a) )^2 = a

x=18+12sqrt(2)+6sqrt(10)+8sqrt(5)/4-2

Subtract terms

x=18+12sqrt(2)+6sqrt(10)+8sqrt(5)/2

With the denominator as simplified as possible, we can see that there is a common factor of 2 shared between the denominator and each of the terms in the numerator. Let's reduce our fraction by this common factor.

x=18+12sqrt(2)+6sqrt(10)+8sqrt(5)/2

Write as a sum of fractions

x=18/2+12sqrt(2)/2+6sqrt(10)/2+8sqrt(5)/2

Simplify quotient

x=9+6sqrt(2)+3sqrt(10)+4sqrt(5)

We obtained the exact solution to the equation. Finally, let's use a calculator to approximate this solution to the nearest tenth.

x=9+6sqrt(2)+3sqrt(10)+4sqrt(5)

Use a calculator

x=9+6(1.414214...)+3(3.162278...)+4(2.236068...)

▼

Evaluate right-hand side

Multiply

x=9+8.485281...+9.486833...+8.944272...

Add terms

x=35.916386...

Round to 1 decimal place(s)

x≈ 35.9

Subchapter links

Lesson Check p.629

Practice and Problem-Solving Exercises p.629-631

Standardized Test Prep p.631

Mixed Review p.631