Before multiplying the given radical expressions, we need to answer two questions.
- Can the expressions be multiplied?
- If so, do symbols need to be added to the answer?
The rule regarding multiplying states that if and are real numbers, then
Because we are assuming that both radicals are and we can see that the given expressions have the same index, we can multiply them. Now, to answer the second question, consider the rule regarding absolute value symbols.
Since both radicals are real numbers and the roots are even, the expressions underneath the radicals are positive. Otherwise, the radicals would be . With this in mind, let's consider the possible values of the variables, and
- In the first radical, the index is even, the of is odd, and the exponent of is even. Therefore, in order for this radical expression to result in a real number, must be positive.
- In the second radical, the index is even and the exponents of and are odd. Since must be positive, in order for this radical expression to result in a real number, also must be positive.
This means that both variables can only take positive values, so we do not need absolute value symbols.
Next, let's simplify the radical expression by finding all of the perfect
powers inside the radical.
Split into factors and write as powers