For any real number a, we have that sqrt(a^n)=a if n is an odd number, and that sqrt(a^n)=|a| if n is even.
a^2sqrt(5ab)/b^7
Practice makes perfect
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 sqrt(a) and sqrt(b) are real numbers and b≠0, then sqrt(a)÷sqrt(b)=sqrt(a÷ b).
sqrt(5a^5)/sqrt(b^(13))=sqrt(5a^5/b^(13))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.
sqrt(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 positive. Otherwise, the radicals would be imaginary. With this in mind, let's consider the possible values of the variables a and b.
In the numerator, the index is even and the exponent of a is odd. For the expression to result in a real number, the value of a must be positive.
In the denominator, the index is even and the exponent of b is odd. For the expression to result in a real number, the value of b must be positive.
Since both variables can only take positive values, we do not need absolute value symbols.
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 sqrt(b) in the denominator. To avoid this, we can multiply the numerator and denominator by a factor that will create a perfect square, which in this case is b.
