What is the Square Root of Five Times the Square Root of Five?

What is the Square Root of Five Times the Square Root of Five?

The square root of five times the square root of five simplifies to the square root of 25, which is equal to 5. When you multiply the square root of 5 by itself, you get 5, and its square root is the original number, which is 5. So, the square root of five times the square root of five is 5.

Simplifying a Product of Square Roots

When multiplying expressions involving square roots, we can often simplify them by using some key properties. Multiplication of radicals follows certain rules that allow us to simplify the products down to a single term.

Let’s look at simplifying the product √5 * √5.

Understanding Square Roots

A square root of a number n is a value that, when multiplied by itself, equals n. For example:

√9 = 3 because 3 * 3 = 9

So √n * √n = n, based on the definition of a square root.

Multiplying Square Roots

When multiplying square roots with the same radicand (the number under the radical), we can apply the following property:

√a * √a = a

In our example, we have:

√5 * √5

Both radicals have a radicand of 5. Therefore, using the multiplying property:

√5 * √5 = 5

The square root expressions simplify to just the radicand by itself.

Explanation

To understand why this works, consider:

√5 * √5 = (√5)*(√5)
= √(5 * 5)
= √25 = 5

Breaking it down shows how the product of square roots of equal radicands produces the radicand itself.

This simplification makes multiplying square root expressions much easier. Understanding the logic behind it builds algebraic skills and conceptual understanding.

In summary, when multiplying square roots with the same radicand, they can be simplified to just the radicand itself following the rule:

√a * √a = a