Square Root Approximation $$ \sqrt{99} $$

Square Root Approximation

In this lesson, we learn how to estimate the value of a square root using a simple and effective approximation method. Our focus is to find the approximate value of the square root of 99 without using a calculator.

1. Theory of Square Root Approximation

Square root approximation is a technique used to estimate the value of a square root when the number is not a perfect square. This method is especially useful for quick calculations in mathematics, science, and engineering.

2. Concept Explanation

The key idea is to use the nearest perfect square to the given number. By comparing the number with a nearby perfect square, we can estimate the square root using a simple formula.

3. Steps or Formula

The approximation formula used is:

$$ \sqrt{a} \approx \frac{a + b}{2\sqrt{b}} $$

where:

• a is the number whose square root we want to estimate
• b is the nearest perfect square to a

4. Example Problem

Estimate the value of:

$$ \sqrt{99} $$

Step 1: Identify values.

The nearest perfect square to 99 is 100.

$$ a = 99, \quad b = 100 $$

Step 2: Substitute into the formula.

$$ \sqrt{99} \approx \frac{99 + 100}{2\sqrt{100}} $$

Step 3: Simplify the expression.

$$ \sqrt{100} = 10 $$ $$ \frac{199}{2 \times 10} = \frac{199}{20} $$ $$ = 9.95 $$

5. Final Answer

The approximate value of \(\sqrt{99}\) is 9.95.

This estimate is very close to the actual value of approximately 9.94987, making this method highly accurate. Square root approximation is useful for fast calculations, estimating dimensions, and solving practical problems when exact values are not required.