Problem 2 - Algebra 2

Question

Find all the real square roots of each number. $$ 0.0049 $$

Step-by-Step Solution

Answer

\(\pm0.07\)

Step 1: Identify the Square Root

To find the square root of 0.0049, we need to look for a number that when multiplied by itself gives the original number, 0.0049.

Step 2: Calculate the Square Root

The square root of a decimal can be found by converting it into fraction form or by using a calculator. The square root of 0.0049 is 0.07 because 0.07 times 0.07 equals 0.0049.

Step 3: Consider Both Positives and Negatives

When finding the real square roots, we must consider both the positive and negative roots because both \(0.07\times0.07 = 0.0049\) and \(\-0.07\times\-0.07 = 0.0049\).

Key Concepts

Square Root Calculation

Understanding how to calculate square roots is crucial when it comes to solving various mathematical problems. The square root of a number is a value that, when multiplied by itself, yields the original number. For example, the square root of 9 is 3, as 3 times 3 equals 9.

When dealing with perfect squares like 9, 16 or 25, finding the square root is straightforward. However, with non-perfect squares or decimals such as 0.0049, the process may seem a bit daunting, but it can be simplified.

In order to calculate the square root of a decimal like 0.0049, you may:

• Convert it to a fraction and simplify,
• Use long division,
• Or employ a calculator for quick and accurate results.

For 0.0049, either approach will give us 0.07 as the square root because multiplying 0.07 by itself results in the original decimal. Hence, recognizing the number which squared gives back the original number, is at the heart of square root calculation.

Positive and Negative Roots

A commonly overlooked aspect when calculating square roots is the existence of both positive and negative roots. Since squaring a negative number results in a positive number—for instance, \( -3 \times -3 = 9 \)—it implies that both positive and negative numbers can be square roots of a given positive number.

Take the number 0.0049; we found its square root to be 0.07. But it’s important to remember that -0.07 is also a square root because \( -0.07 \times -0.07 = 0.0049 \). Every positive real number has two real square roots, one positive and one negative. This principle should be applied whenever you're asked to 'Find all the real square roots' of a number, as these tasks require you to consider both possibilities.

Solving Decimal Squares

Working with decimal squares might seem more complex than dealing with whole numbers, but the principles remain the same. When solving for the square roots of decimal numbers, the process may involve estimation, fraction conversion, or the use of digital tools such as calculators.

The key steps to solve decimal squares effectively are:

• Estimate the square root to get an initial idea of the number range,
• Perfect the square root using long division if necessary,
• Confirm the result using a calculator,
• Consider both the positive and negative square roots.

By understanding these steps, and the fact that decimals result from dividing whole numbers, you'll be equipped to address square roots of decimal numbers with confidence.

Solving decimal squares helps in understanding the behavior of squares and square roots in a more nuanced way, which further applies to numerous real-world applications in science, engineering, and finance.