The Correct Answer and Explanation is:
The correct answer is -4√5.
This problem asks you to combine two radical expressions: -√5 – 3√5. The key to solving this is understanding the concept of “like radicals.” Like radicals are terms that have the exact same radical part, which means the number inside the radical sign (the radicand) and the type of root (the index) are identical. In this case, both terms, -√5 and -3√5, are square roots and both have a radicand of 5. This makes them like radicals, and they can be combined through addition or subtraction.
Think of this process as being very similar to combining like terms in algebra. For example, if you were asked to simplify the expression -x – 3x, you would combine the coefficients (the numbers in front of the variable x). The term -x has an implied coefficient of -1. So, you would calculate -1 – 3, which equals -4, and keep the variable x. The result would be -4x.
We apply the exact same logic to our radical expression. The first term, -√5, can be written as -1√5. The second term is -3√5. To combine them, we focus on their coefficients, which are -1 and -3.
We perform the operation on the coefficients:
-1 – 3 = -4
After combining the coefficients, we keep the common radical part, which is √5, unchanged. We do not add or subtract the numbers inside the radicals.
Therefore, combining the new coefficient (-4) with the original radical part (√5) gives us the final answer: -4√5.
The other options are incorrect because they either miscalculate the coefficient or incorrectly alter the radicand. You cannot add the radicands (5 + 5) to get √10. The radical part must remain the same when combining like terms.
