Problem 25 - 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 result is -8.

Step 1: Identify the Square Root

First, we need to identify the square root operation within the problem. The problem asks for \(-\sqrt{64}\). This means we first need to calculate \(\sqrt{64}\), which is asking what number multiplied by itself gives 64.

Step 2: Calculate the Square Root

Calculate \(\sqrt{64}\). Since 8 multiplied by itself equals 64 (i.e., \(8 \times 8 = 64\)), it follows that \(\sqrt{64} = 8\).

Step 3: Apply the Negative Sign

The original problem was \(-\sqrt{64}\). Since we found that \(\sqrt{64} = 8\), we now need to apply the negative sign as indicated in the problem. This results in \(-8\).

Key Concepts

Understanding Square Root Calculation

Square root calculation is a fundamental math concept that often confuses students. **Square roots** are essentially the numbers that, when multiplied by themselves, yield another number. For example, the square root of 64 is a calculation that searches for a number which, when squared, results in 64. Many times, the process starts with identifying perfect squares, numbers which are products of an integer multiplied by itself. In this case, 64 is a perfect square because it equals 8 times 8.

• To calculate a square root manually, you determine which number squared gives you the original number.
• For 64, since 8 x 8 = 64, the square root of 64 is 8.

Learning to identify perfect squares like 4, 16, 25, 36, 49, 64, 81, and 100 can be very helpful in quickly finding square roots.

Exploring Negative Square Roots

A common point of confusion arises with **negative square roots**. When you see a sign in front of a square root, it changes the calculation slightly. For example, in \(-\sqrt{64}\), you're asked to first find the square root of 64, which is 8. But what differentiates this calculation is the negative sign outside the square root.

• This sign indicates that once you calculate the square root, you need to make the result negative.
• In a practical sense, this changes the final answer from 8 to -8.

The negative square root operation does not imply finding a square root of a negative number (which is not a real number), but rather taking the negative of the resulting positive square root.

Manual Computation Techniques for Square Roots

While calculators provide a fast way to compute square roots, knowing how to proceed manually is useful for understanding the workings behind the operations. **Manual computation techniques** involve estimating, squaring, and adjusting to reach the desired number. Here's how you can compute a square root manually:

• Start by estimating a number close to your original number that is easier to handle. For instance, for 64, start with numbers like 5 or 10 as trial estimates.
• Square your trial numbers and see which squares come closer to your original number (in this case, 64).
• If your initial estimates are far off, adjust them based on whether the square was too low or too high. Fine-tuning your estimate helps you identify the square root.

Practice with these techniques reinforces understanding and familiarization with square roots, benefiting both computational skills and mathematical intuition.