Problem 64 - Prealgebra

Question

Estimate each square root to the nearest whole number. Do not use a calculator. $$\sqrt{8.67}$$

Step-by-Step Solution

Answer

The estimated square root of 8.67 is approximately 3.

Step 1: Identify Perfect Squares

Identify the perfect squares around 8.67. The perfect squares close to 8.67 are 4 (since \(2^2 = 4\)) and 9 (since \(3^2 = 9\)).

Step 2: Find the Nearest Perfect Squares

Determine which perfect square \(8.67\) is nearer to. \(8.67\) is closer to \(9\) than it is to \(4\) because 8.67 is only 0.33 away from 9 and 4.67 away from 4.

Step 3: Determine the Nearest Whole Number

Since \(8.67\) is closer to \(9\) than it is to \(4\), the square root of \(9\) is a better estimate for \(\sqrt{8.67}\) than the square root of \(4\). Thus, the nearest whole number is 3.

Key Concepts

Perfect Squares

Perfect squares are numbers that result from squaring a whole number. In simple terms, if you multiply a number by itself and it gives you a whole number, then that whole number is a perfect square. For instance, 4 is a perfect square because it equals \(2 \times 2\), and 9 is a perfect square because it equals \(3 \times 3\).

The importance of perfect squares in calculating or estimating square roots lies in their stability. Recognizing perfect squares around your target number provides you with benchmarks. This helps in understanding where your number falls concerning these stable values.

• For smaller numbers like \(\sqrt{8.67}\), perfect squares nearby (like 4 and 9) tell us that 8.67 is more like 9 and less like 4.
• This concept is essential because it gives you a direct method to estimate square roots without intense calculations or tools.

Whole Numbers

Whole numbers are the set of natural numbers including zero. They are numbers without fractions or decimals. For example, \(0, 1, 2, 3, 4\), and so on, are whole numbers.

When estimating square roots, finding the nearest whole number is crucial. This is because square roots can often be messy decimal numbers, which are more challenging to quickly compute.

• By considering the whole numbers around your target square root, you simplify the problem enormously.
• In the example of estimating \(\sqrt{8.67}\), the answer approximates to a clean, easy number - 3 - which is a whole number.

This approximation helps to bring clarity and understanding, particularly when precision in calculations is less critical than speed or ease.

Mathematical Estimation

Mathematical estimation involves making an educated guess about the value of a mathematical expression. This approach speeds up problem-solving without sacrificing accuracy significantly. When estimating square roots, like with \(\sqrt{8.67}\), the goal is to figure out which whole number the root is closest to by comparing it to nearby perfect squares.

Estimation relies on understanding and manipulation of basic mathematical properties. In our case, it involves comparing differences:

• Since \(8.67\) is closer to 9 than it is to 4, \(\sqrt{8.67}\) is closer to 3 than it is to 2.
• This process is beneficial because it can be performed quickly in your head, especially when you don't have a calculator at hand.

Mathematical estimation therefore, isn't about exactness. Instead, it's about finding a result that is practical and sufficient for most everyday purposes without unnecessary complexity.