Problem 46 - Prealgebra

Question

Compute the exact square root. \(\sqrt{4.41}\)

Step-by-Step Solution

Answer

The exact square root of 4.41 is 2.1.

Step 1: Understand the Problem

We need to compute the exact square root of 4.41. The goal is to find a number which, when multiplied by itself, equals 4.41.

Step 2: Express in Decimal Form

Recognize that 4.41 can be expressed in decimal form as 4.41 = 441/100.

Step 3: Simplify the Fraction

Compute the square root by breaking down the fraction: \[ \sqrt{\frac{441}{100}} = \frac{\sqrt{441}}{\sqrt{100}} \]

Step 4: Calculate the Square Roots

Identify \( \sqrt{441} \) and \( \sqrt{100} \). 441 is a perfect square that equals 21 * 21, so \( \sqrt{441} = 21 \). 100 is a perfect square that equals 10 * 10, so \( \sqrt{100} = 10 \).

Step 5: Divide the Simplified Roots

Divide the resulting roots to find the final answer: \[ \frac{21}{10} = 2.1 \]

Step 6: Confirm Calculation

To ensure accuracy, square the found root: \( 2.1 \times 2.1 = 4.41 \), which matches the initial number.

Key Concepts

Decimal Form

Decimals represent numbers as a fraction of a whole, using powers of ten. In this exercise, the number 4.41 readily appears in decimal form. The key is understanding how it transitions to a fraction. The decimal "4.41" indicates 4 whole units and 41 hundredths, as the digit "41" is two places past the decimal point. This decimal converts to the fraction \( \frac{441}{100} \) because the place value after the decimal goes up to the hundredth place. Understanding decimal forms as fractions is crucial when working with square roots, as it facilitates easier calculations. Converting decimals into fractions can help simplify calculations, ensuring that operations like finding square roots don’t seem daunting. The transformation from decimal to fraction involves multiplying the decimal value by a power of 10 (specifically the place value). In the 4.41 example, this involves moving the decimal two places (multiplying and dividing by 100). Knowing how to maneuver between these forms can deeply enhance understanding and efficiency in math problems.

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. Recognizing perfect squares provides a shortcut to finding square roots.For example, in our exercise, the number 441 is identified as a perfect square. Perfect squares simplify the calculation of square roots since the root results in an integer — a straightforward and neat outcome. Other common perfect squares include:

• 1 (as \(1 \times 1 = 1\))
• 4 (as \(2 \times 2 = 4\))
• 9 (as \(3 \times 3 = 9\))
• 16 (as \(4 \times 4 = 16\))

By familiarizing yourself with perfect squares, you increase your ability to quickly recognize and solve square root problems. Knowing these can save time and enhance accuracy when calculating or simplifying radical expressions.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form, making calculations more manageable and results easier to interpret.In this particular exercise, after recognizing that the square root of 441 is 21 and 100 is 10, the fraction \( \frac{21}{10} \) emerges from dividing these results. In some cases, you may need to check if a fraction can be further reduced by finding the greatest common divisor. However, with \( \frac{21}{10} \), it’s already in its simplest form since 21 and 10 share no common divisors other than 1.Fraction simplification is vital in mathematical processes, ensuring clarity and accuracy. It leads you efficiently to an answer, whether presented in decimal form or left as a fraction. Also, understanding the simplification process strengthens fundamental math skills, laying down a strong foundation for solving more complex math problems.