Exercise 34 Page 248

Let's start by considering the given expression. sqrt(4c^3d^3)* sqrt(8c^3d) Note that the variables c and d have an odd exponent in both radicals. Therefore, for the expression to be defined, these variables must have the same sign — both positive or both negative. This means that we will need absolute value symbols when extracting the variables from the radicals. Let's start by recalling the Product Property of Radicals. sqrt(a)* sqrt(b)=sqrt(a* b), fora≥0,b≥0 We will use this property to simplify our expression.

sqrt(4c^3d^3)* sqrt(8c^3d)

ProdSqrt

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

sqrt(4c^3d^3(8)c^3d)

CommutativePropMult

Commutative Property of Multiplication

sqrt(4(8)c^3c^3dd^3)

Multiply

Multiply

sqrt(32c^3c^3dd^3)

MultPow

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

sqrt(32c^6dd^3)

MultBasePow

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

sqrt(32c^6d^4)

SplitIntoFactors

Split into factors

sqrt(16(2)c^(3(2))d^(2(2)))

WritePow

Write as a power

sqrt(4^2(2)c^(3(2))d^(2(2)))

ProdInExponent

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

sqrt(4^2(2)(c^3)^2(d^2)^2)

Now that we have simplified the radicand 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(4^2(2)(c^3)^2(d^2)^2)

SqrtProd

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

sqrt(4^2)sqrt(2)sqrt((c^3)^2)sqrt((d^2)^2)

SqrtPowToNumber

sqrt(a^2)=a

4sqrt(2)sqrt((c^3)^2)sqrt((d^2)^2)

SqrtToAbs

sqrt(a^2)=|a|

4sqrt(2)|c^3||d^2|

CommutativePropMult

Commutative Property of Multiplication

4|c^3||d^2|sqrt(2)

Finally, since d^2 is always non-negative, we can remove the absolute value symbol from this expression. 4|c^3||d^2|sqrt(2) ⇔ 4|c^3|d^2sqrt(2)