Problem 40 - Prealgebra

Question

Compute the exact square root. \(\sqrt{5.76}\)

Step-by-Step Solution

Answer

The exact square root of 5.76 is 2.4.

Step 1: Estimate the square root

First, try to estimate the square root by finding two perfect squares between which 5.76 falls. We know that 2 squared is 4 and 3 squared is 9. Since 5.76 is between 4 and 9, the square root is between 2 and 3.

Step 2: Consider Decimal Values

Given that 5.76 is much closer to 4 than it is to 9, let's estimate that the square root is closer to 2.4 because 2.4 squared is equal to 5.76.

Step 3: Verify Using Multiplication

Multiply 2.4 by itself to check if the result equals 5.76. Calculate: \(2.4 imes 2.4 = 5.76\).Since it equals 5.76, then the square root of 5.76 is indeed 2.4.

Key Concepts

Estimation

Estimating square roots is a helpful way to get a closer idea of what the answer may be before diving into more precise methods. This step can be visualized as locating two perfect squares between which our number falls.

In the exercise given, we start by identifying that 5.76 lies between two smaller perfect squares, 4 and 9. The square roots of these values are 2 and 3 respectively. Thus, \( \sqrt{5.76} \ \) should logically be a value somewhere between 2 and 3. Estimation gives us that vital first step to understanding how close we might be to our final answer.

• Identify the closest perfect squares that are smaller and larger than the given number.
• Use these squares to roughly guess a possible range for the square root.

Estimating does not give us the exact square root, but it sets up the path we need to follow to reach a more precise answer.

Decimal Numbers

Working with decimal numbers, like 5.76, requires us to think a bit more about precision and where these numbers lie on the number line. In our example, the number 5.76 is closer to 4 than it is to 9, suggesting that the square root will lean more towards 2 than 3.

When dealing with decimals:

• Break them down if possible into familiar components (like recognizing that 0.76 is nearing the halfway point between 0 and 1).
• Consider estimating decimals by incrementally evaluating them (such as using 2.1, 2.2, 2.3, etc.).

In this scenario, considering decimal differences helps, where trying values such as 2.4 shows potential. Calculating \( 2.4 \times 2.4 = 5.76 \ \) verifies our estimation, showing precision in decimals not just as simple counts, but valuable tools in exact sciences.

Multiplication Verification

Verifying your result using multiplication is a crucial step in mathematics. Once you think you have the correct square root, checking if your squaring returns the original number is the perfect method to confirm. This can save you from potentially overlooking a mistake.

In our exercise, after estimating that \(\sqrt{5.76} \approx 2.4\), we need to verify: \Calculating \(2.4 \times 2.4\) results in 5.76. This matches the original number given, thereby confirming our answer.

• Multiply your estimated square root by itself.
• Check if this multiplication gives you the original number you started with.

This verification act is a logical step, ensuring that our process is solid and our answer reliable. It reinforces confidence in the solution and strengthens calculation skills overall.