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Pearson Algebra 1 Common Core, 2011

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Pearson Algebra 1 Common Core, 2011 View details

Mid-Chapter Quiz

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Exercise 30 Page 632

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

Exact Solution: 6sqrt(10)-6sqrt(5)
Approximated Solution: 5.6

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.

x/sqrt(10)=3sqrt(2)/sqrt(2)+1

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

x=3sqrt(2)/sqrt(2)+1* sqrt(10)

MoveRightFacToNum

a/c* b = a* b/c

x=3sqrt(2)* sqrt(10)/sqrt(2)+1

SplitIntoFactors

Split into factors

x=3sqrt(2)* sqrt(2* 5)/sqrt(2)+1

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

x=3sqrt(2)* sqrt(2)* sqrt(5)/sqrt(2)+1

sqrt(a)* sqrt(a)= a

x=3* 2* sqrt(5)/sqrt(2)+1

Multiply

x=6sqrt(5)/sqrt(2)+1

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 second term of the expression. Denominator:& sqrt(2) + 1 Conjugate:& sqrt(2) - 1 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=6sqrt(5)/sqrt(2)+1 ⇕ x=6sqrt(5)/sqrt(2)+1* sqrt(2)-1/sqrt(2)-1 We can simplify the right-hand side to find the value of x. Let's first simplify the numerator.

x=6sqrt(5)/sqrt(2)+1* sqrt(2)-1/sqrt(2)-1

Multiply fractions

x=(6sqrt(5))(sqrt(2)-1)/(sqrt(2)+1)(sqrt(2)-1)

▼

Multiply

Distribute 6sqrt(5)

x=6sqrt(5)* sqrt(2)-6sqrt(5)* 1/(sqrt(2)+1)(sqrt(2)-1)

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

x=6sqrt(10)-6sqrt(5)* 1/(sqrt(2)+1)(sqrt(2)-1)

Identity Property of Multiplication

x=6sqrt(10)-6sqrt(5)/(sqrt(2)+1)(sqrt(2)-1)

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

x=6sqrt(10)-6sqrt(5)/(sqrt(2)+1)(sqrt(2)-1)

ExpandDiffSquares

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

x=6sqrt(10)-6sqrt(5)/(sqrt(2))^2-(1)^2

▼

Simplify

1^a=1

x=6sqrt(10)-6sqrt(5)/(sqrt(2))^2-1

( sqrt(a) )^2 = a

x=6sqrt(10)-6sqrt(5)/2-1

Subtract term

x=6sqrt(10)-6sqrt(5)/1

a/1=a

x=6sqrt(10)-6sqrt(5)

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

x=6sqrt(10)-6sqrt(5)

Use a calculator

x=6(3.162278...)-6(2.236068...)

▼

Evaluate right-hand side

Multiply

x=18.973666...-13.416407...

Subtract term

x=5.557259...

Round to 1 decimal place(s)

x≈ 5.6