Exercise 19 Page 265

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 a negative value — the range of a square root function is all real numbers greater than or equal to 0. Now, consider the given radical expression. sqrt(36x^2y^7) The exponent of x is even and, in the simplified expression, this variable will have an odd exponent. Therefore, when we remove x from the radical, we will 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(36x^2y^7)

WritePow

Write as a power

sqrt(6^2x^2y^7)

WriteSum

Write as a sum

sqrt(6^2x^2y^(1+6))

SumInExponentOne

a^(1+m)=a*a^m

sqrt(6^2x^2yy^6)

CommutativePropMult

Commutative Property of Multiplication

sqrt(6^2x^2y^6y)

SplitIntoFactors

Split into factors

sqrt(6^2x^2y^(3(2))y)

ProdInExponent

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

sqrt(6^2x^2(y^3)^2y)

SqrtProd

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

sqrt(6^2)sqrt(x^2)sqrt((y^3)^2)sqrt(y)

SqrtPowToNumber

sqrt(a^2)=a

6sqrt(x^2)y^3sqrt(y)

SqrtToAbs

sqrt(a^2)=|a|

6|x|y^3sqrt(y)