Problem 65 - Prealgebra

Question

Estimate each square root to the nearest whole number. Do not use a calculator. $$-\sqrt{19.85}$$

Step-by-Step Solution

Answer

The neatest whole number estimate for \(-\sqrt{19.85}\) is \(-4\).

Step 1: Identify Perfect Squares Around

First, identify the two perfect squares nearest to 19.85. The perfect squares are 16 and 25, because \(16 = 4^2\) and \(25 = 5^2\). Thus, we know that \(4 < \sqrt{19.85} < 5\).

Step 2: Estimate the Square Root

Since 19.85 is closer to 16 than to 25, we estimate \( \sqrt{19.85} \approx 4\). The gap is larger between 19.85 and 25 than between 19.85 and 16.

Step 3: Apply Negation

The original problem is to calculate \(-\sqrt{19.85}\). Since \( \sqrt{19.85} \approx 4\), apply the negation to obtain \(-4\).

Key Concepts

Perfect Squares

In mathematics, perfect squares play a crucial role in understanding square roots. A perfect square is a number that can be expressed as the product of an integer multiplied by itself. For example, the number 16 is a perfect square because it equals \(4 \times 4\). The series of perfect squares begins with numbers like 1, 4, 9, 16, 25, and so on.

• Understanding Perfect Squares: Perfect squares are easy to identify because they result from squaring whole numbers. This makes them useful as reference points when estimating other square roots.
• Examples: \(49 = 7^2\), \(81 = 9^2\), and \(100 = 10^2\).

Identifying perfect squares near a given number can help you approximate the square root of non-perfect squares. In the exercise, the perfect squares closest to 19.85 are 16 (\(4^2\)) and 25 (\(5^2\)). Knowing these references helps to form a boundary for estimation.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. Finding the square root of perfect squares is straightforward. For non-perfect squares, like 19.85, estimation between known perfect square roots is necessary.

• Square Root Function: The symbol \(\sqrt{}\) denotes the square root. For example, \(\sqrt{16} = 4\) because 4 is the number that squared results in 16.
• Estimation Technique: By observing perfect squares on either side of 19.85, one can deduce that \(4 < \sqrt{19.85} < 5\). Since 19.85 is closer to 16 than to 25, \(\sqrt{19.85}\) is estimated to be approximately 4.

This method of estimation can be helpful in determining unknown square roots without a calculator, making mental math simpler and quicker for learners.

Negation in Mathematics

Negation in mathematics refers to applying a negative sign to a value or expression, effectively reversing its sign. It plays a significant role in various calculations, including square roots.

• Understanding Negation: The negative sign (\(-\)) changes a positive number to its negative counterpart. In mathematics, handling negative values correctly is essential for accurate results.
• Application to Square Roots: In the case of estimating \(-\sqrt{19.85}\), once we estimate \(\sqrt{19.85} \approx 4\), applying the negation results in \(-4\). This accurately reflects the original problem involving a negative square root.

The exercise demonstrates negation's application in practical problems, ensuring students understand its effects both conceptually and in calculations.