Problem 34 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Check the results by squaring each root. $$ -\sqrt{225} $$

Step-by-Step Solution

Answer

The expression \(-\sqrt{225}\) evaluates to \(-15\). Verification by squaring proves correct, as \((-15)^2 = 225\).

Step 1: Evaluate the Square Root

First, it is necessary to evaluate the square root without considering the negative sign. The square root of 225 is either \(+15\) or \(-15\). In this case, because there is a negative sign in front of the square root, we take \(-15\). The expression \(-\sqrt{225}\) thus evaluates to \(-15\).

Step 2: Verify the Result by Squaring

To check our result, we square \(-15\). That is, we calculate \((-15)^2\). Squaring a negative number always results in a positive value, so \((-15)^2 = 225\).

Key Concepts

Square Root Calculation

Understanding how to calculate a square root is essential in various fields of mathematics and science. A square root of a number 'x' is a value 'y' such that y^2 = x. To find the square root of 225, as illustrated in the exercise, you look for a positive number that, when multiplied by itself, gives you 225. This process can be done by factoring, using a calculator, or remembering common square numbers.

When you calculate the square root of 225, you get 15, because 15^2 = 225. It's important to note that although square roots typically yield a positive and a negative result, by convention, the term 'the square root' refers to the principal, or non-negative, square root. However, when a negative sign is explicitly placed before the square root symbol, as is the case in our exercise -25, the negative result is taken into account, leading us to -15.

Negative Numbers in Square Roots

The concept of square roots becomes slightly more complex with the introduction of negative numbers. The square root of a positive number can have both a positive and negative value since squaring either yields the original positive number. However, it's essential to distinguish that the square root itself represents only the positive value. When it comes to negative numbers under a square root sign (i.e., √-x), it implies the realm of complex numbers, as no real number squared will give a negative result. The solution to such an expression is an imaginary number.

The exercise given poses a different situation: -25 and not √-225. Here, the negative sign is outside of the square root, which dictates that we first find the square root of the positive number 225 and then apply the negative sign, yielding -15.

Verifying Square Root Solutions

Verifying your solution is an excellent practice to ensure you've calculated the square root correctly. As shown in the textbook exercise, after evaluating the square root, you can verify it by squaring the result. If the square of your result equals the original number inside the square root sign, you've found the correct square root value.

In the context of the exercise, squaring -15 results in (-15)^2 = 225. This positive outcome confirms the solution, as the negative sign is squared along with the number, which always results in a positive product. This step not only verifies the square root but also helps to understand how the square of a negative number functions – it negates the negative sign, thus reinforcing the rule that the square of any real number cannot be negative.