Exercise 42 Page 371

Before dividing the given radical expressions, we need to answer two questions.

Can the expressions be divided?
If so, do absolute value symbols need to be added to the answer?

The rule regarding dividing radical expressions states that if

n a

and

n b

are real numbers and

b \neq = 0,

then

n a \div n b = n a \div b .

\frac{2 0 a b}{4 5 a ^{2} b ^{3}} = \frac{2 0 a b}{4 5 a ^{2} b ^{3}}

Because we are assuming that both radicals are real numbers and we can see that the given expressions have the same index, we can divide them. Now, to answer the second question, consider the rule regarding absolute value symbols for any real number

a .

n a^{n} = {∣ a ∣ if n is odd ∣ a ∣ if n is even

Since both radicals are real numbers and the roots are even, the expressions underneath the radicals are non-negative. Otherwise, the radicals would be imaginary. With this in mind, let's consider the possible values of the variables

a

and

b .

Since both $a$ and $b$ appear in the denominator, they cannot be equal to $0 .$
In the numerator, the index is even and the exponents of $a$ and $b$ are odd. For the expression to result in a real number, the product of $a$ and $b$ must be positive. Therefore, $a$ and $b$ must have the same sign — both positive or both negative.
In the denominator, the index is even so the expression contained in the radical must be positive.
- The product of $a^{2}$ and $b^{3}$ must be positive — $a^{2}$ and $b^{3}$ must have the same sign.
- Because the result of $a^{2}$ will always be positive, $b^{3}$ must also be positive.
If $b^{3}$ is positive, $b$ is positive. Since $a$ and $b$ must have the same sign, $a$ and $b$ must both be positive.

Since both variables can only take positive values, we do not need absolute value symbols.

\frac{2 0 a b}{4 5 a ^{2} b ^{3}}

DivSqrt

$\frac{a}{b} = \frac{a}{b}$

\frac{2 0 a b}{4 5 a ^{2} b ^{3}}

ReduceFrac

$\frac{a}{b} = \frac{a / 5 a b}{b / 5 a b}$

\frac{4}{9 a b ^{2}}

Next, let's simplify the radical expression by finding all of the perfect squares inside the radical.

\frac{4}{9 a b ^{2}}

WritePow

Write as a power

\frac{2 ^{2}}{3 ^{2} a b ^{2}}

CommutativePropMult

Commutative Property of Multiplication

\frac{2 ^{2}}{3 ^{2} b ^{2} a}

Let's stop here for a moment and consider the fact that we need to have a rationalized denominator. If we simplified all of the perfect squares as is, we would be left with