Exercise 7 Page 248

Before we attempt to simplify the given radical expression, let's consider one of the more difficult parts of correctly simplifying radicals. When the exponent of a variable inside the radical is even and the simplified expression for that variable has an odd exponent, we need to use absolute value symbols. ll sqrt(x^2)= |x| & sqrt(x^3)=x sqrt(x) sqrt(x^4)=x^2 & sqrt(x^6)=|x^3| This is because we do not want the result to be negative — the range of a square root function is all real numbers greater than or equal to 0. Now, consider the given radical expression. sqrt(60x^4y^7) The exponent of x is even and, in the simplified expression, this variable will have an even exponent. Therefore, when we remove x from the radical, we will not need absolute value symbols. The exponent of y is odd, so we will not need absolute value symbols when we remove y from the radical.

sqrt(60x^4y^7)

SplitIntoFactors

Split into factors

sqrt(15* 4x^4y^7)

WritePow

Write as a power

sqrt(15*2^2x^4y^7)

WriteSum

Write as a sum

sqrt(15*2^2x^4y^((6+1)))

SumInExponent

a^(m+n)=a^m*a^n

sqrt(15*2^2x^4y^6y^1)

ExponentOne

a^1=a

sqrt(15*2^2x^4y^6y)

Now that we have simplified the factors as much as possible, let's recall the Product Property of Square Roots. sqrt(a* b)=sqrt(a) * sqrt(b), for a≥ 0,b≥ 0 Let's use this property to simplify our expression even further.

sqrt(15*2^2x^4y^6y)

SqrtProd

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

sqrt(15)* sqrt(2^2)*sqrt(x^4)*sqrt(y^6)*sqrt(y)

SplitIntoFactors

Split into factors

sqrt(15)* sqrt(2^2)*sqrt(x^(2(2)))*sqrt(y^(3(2)))*sqrt(y)

ProdInExponent

a^(m* n)=(a^m)^n

sqrt(15)* sqrt(2^2)*sqrt((x^2)^2)*sqrt((y^3)^2)*sqrt(y)

SqrtPowToNumber

sqrt(a^2)=a

sqrt(15)* 2* x^2* y^3* sqrt(y)

CommutativePropMult

Commutative Property of Multiplication

2x^2y^3sqrt(15)sqrt(y)

ProdSqrt

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

2x^2y^3sqrt(15y)