Problem 45 - Prealgebra

Question

Compute the exact square root. \(\sqrt{2.89}\)

Step-by-Step Solution

Answer

The exact square root of 2.89 is 1.7.

Step 1: Recognize the Perfect Square

We start by determining if 2.89 is a perfect square. The number 2.89 can be expressed as a decimal, which we suspect could come from a fractional representation.

Step 2: Convert the Decimal to a Fraction

Rewriting 2.89 as a fraction gives us \(\frac{289}{100}\). This conversion is done by recognizing that 2.89 is equivalent to 289 divided by 100.

Step 3: Find the Square Root of the Fraction

To find \(\sqrt{\frac{289}{100}}\), we apply the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). Thus, we compute \(\sqrt{289}\) and \(\sqrt{100}\).

Step 4: Compute \(\sqrt{289}\) and \(\sqrt{100}\)

We recognize \(\sqrt{289} = 17\) because \(17^2 = 289\). Similarly, \(\sqrt{100} = 10\) because \(10^2 = 100\).

Step 5: Simplify the Result

Using the values from Step 4, we find \(\sqrt{\frac{289}{100}} = \frac{17}{10} = 1.7\). Thus, the exact square root of 2.89 is 1.7.

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. In simpler terms, when you multiply an integer by itself, you get a perfect square. For example, the number 16 is a perfect square because it equals 4 multiplied by 4. The concept of perfect squares is crucial when solving square root problems, as it helps identify numbers whose square roots are integers. Perfect squares are often memorized because they frequently appear in mathematical problems and can simplify calculations.

• Examples of perfect squares are 1, 4, 9, 16, 25, and so on.
• Recognizing perfect squares quickly helps in mental math and algebraic manipulations.
• Identifying non-perfect squares can guide you in approximating roots or converting them to fractions.

In the context of our exercise, recognizing whether 2.89 is a perfect square helps determine its potential square root without computational effort.

Fractional Representation

Fractional representation refers to expressing a decimal number in the form of a fraction. This involves writing the number as a ratio of two integers, which is often a very helpful step in solving various mathematical problems. Understanding fractional representation is essential because it can transform complex decimal problems into simpler forms that are easier to manipulate.

• A decimal like 0.5 can be expressed as the fraction \( \frac{1}{2} \).
• More complex decimals, like 2.89, can be converted to \( \frac{289}{100} \) by recognizing the decimal places and using powers of 10.
• Once a decimal is a fraction, other operations, like finding square roots, become more straightforward.

For instance, in our exercise, rewriting 2.89 as \( \frac{289}{100} \) was a critical step in finding its exact square root efficiently.

Decimal to Fraction Conversion

Converting a decimal to a fraction is a key skill in mathematics that makes manipulating and understanding numbers more intuitive. This process involves using the place value of the decimal to transition it into a fraction format, which can then be simplified if necessary. Let's break down converting 2.89:

Steps to Conversion

First, identify the decimal. Each digit after the decimal point increases the place value from tenths to hundredths, and so on. For 2.89:

• The 2 is in the units place.
• The 8 is in the tenths place.
• The 9 is in the hundredths place.

Thus, 2.89 can be expressed as \( \frac{289}{100} \) because you move the decimal two places to the right, turning it into an integer divided by 100.

The Importance of Simplification

Once expressed as a fraction, you may need to simplify it. For our example, \( \frac{289}{100} \) is already in simplest form and useful for operations like finding the square root.Mastering this conversion allows you to transition smoothly between decimals and fractions, a vital skill in both elementary and advanced math contexts.