Problem 25 - Elementary and Intermediate Algebra

Question

Evaluate each square root without using a calculator. See Objective 1 and Example 1. $$ -\sqrt{64} $$

Step-by-Step Solution

Answer

The answer is -8.

Step 1: Identify the Square Root

The problem asks us to evaluate \(-\sqrt{64}\). First, we recognize that \(\sqrt{64}\) asks for the positive number that, when squared, equals 64.

Step 2: Calculate the Square Root

We know from multiplication facts that \(8 \times 8 = 64\). Therefore, the square root of 64 is 8, because 8 squared yields 64.

Step 3: Apply the Negative Sign

The problem includes a negative sign in front of the square root: \(-\sqrt{64}\). Since we've determined that \(\sqrt{64} = 8\), we apply the negative sign to get \(-8\).

Key Concepts

Calculating Square Roots

Calculating square roots is the process of finding a number that, when multiplied by itself, equals the given number. For instance, if you want to find the square root of 64, you're essentially looking for a number that when squared, results in 64.

To identify the square root of a number, you can use multiplication facts or even trial and error with smaller numbers:

• Check if the number is a perfect square. Perfect squares are numbers like 4, 9, 16, 25, etc., which are squares of whole numbers.
• If it is a perfect square, identify what number it is a square of. For example, 64 is a perfect square because it is 8 squared.

Remember, perfect squares are integral values that make calculating square roots straightforward and manageable. This recognition allows you to avoid lengthy calculations, especially when you need to find square roots manually.

Negative Square Roots

Understanding negative square roots involves recognizing how negative signage interacts with square roots. In mathematics, the square root symbol \(\sqrt{}\) typically refers to the principal (positive) square root. However, if there's a negative sign in front of the square root symbol, like in \(-\sqrt{64}\), it affects only the final result, not the number under the radical.Here's how it works:

• First, calculate the square root of the number under the square root symbol, ignoring the negative sign at first. So, compute \(\sqrt{64} = 8\).
• Once you've identified the square root, simply apply the negative sign to your answer. So, you have: \(-8\).

This distinction is important, as it ensures you clearly differentiate between taking the square root and the presence of a negative sign. The presence of a negative sign outside the square root always results in a negative number.

Manual Calculation Steps

If you're finding square roots without a calculator, manual steps can guide you through. Let's see how it works, especially with perfect squares where manual calculations are simpler.First, identify if the number is a perfect square:

• Look for numbers that are known to be perfect squares (e.g., 4, 9, 16, 25).
• 64 is a perfect square, known easily to most because \(8 \times 8 = 64\).

Once confirmed as a perfect square, find the root by considering possible multiplicands:

• Try squaring integers starting from 1, 2, 3, ... until the squared number equals your original number.
• If trying sequential integers, you'd find 8 works because \(8^2 = 64\).
• These basic multiplication checks can aid in verifying the square root.

Even without technology, checking with small and known perfect squares makes finding square roots less daunting. Practice makes the process even faster and more accurate!