Problem 58 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$0.04$$

Step-by-Step Solution

Answer

The square roots of 0.04 are \(+0.2\) and \(-0.2\).

Step 1: Identify the number and Aim

The number to find the square root of is 0.04. The aim is to find such a number that when it is squared, it gives 0.04.

Step 2: Find the square root

The square root of a non-negative number \(A\) can be positive or negative, and both will be square roots since squaring either one will yield \(A\). So, the square roots of 0.04 are \(+0.2\) and \(-0.2\).

Step 3: Check the results

Now check both results by squaring them. Squaring \(+0.2\) gives \(0.04\), and squaring \(-0.2\) also gives \(0.04\). Therefore, both \(+0.2\) and \(-0.2\) are indeed square roots of \(0.04\).

Key Concepts

Square Root Calculation

Understanding how to calculate the square root of a number is an essential skill in mathematics, which is often used to solve various types of problems. The square root of a number is a value that, when multiplied by itself or 'squared', gives the original number. For instance, finding the square root of 0.04 involves looking for a number that, when squared, results in 0.04.
To calculate this, one might first recognize that squaring a number ending in two decimal places will often result in a number with four decimal places. Thus, the square root of 0.04 is expected to be a number with one decimal place. By trying out different values, or using a calculator, we find that 0.2 fits the criteria since \( 0.2 \times 0.2 = 0.04 \).
Using a calculator for square root calculation is straightforward, but understanding the process can be very useful, especially when calculators are not allowed, such as during certain exams or tasks.

Squaring Numbers

The process of squaring numbers involves multiplying a number by itself. This is represented mathematically as \( n^2 \) where \( n \) is any real number. The term 'squared' comes from the geometric concept where a square with sides of length \( n \) has an area of \( n^2 \).

• For example, \( 5^2 = 5 \times 5 = 25 \).
• If we take a decimal number like 0.2, squaring it would be \( 0.2^2 = 0.2 \times 0.2 = 0.04 \) as seen in our exercise.

It's important to note that squaring a positive or negative number will always result in a positive value. This is because the product of two positive or two negative numbers is positive, leading us to our next concept, which involves the implications of this fact for square roots.

Positive and Negative Roots

When finding square roots, it is crucial to recognize that a non-negative number will have two square roots: one positive and one negative. This is because both a positive number and its negative counterpart, when squared, will yield the same positive result. For example, both 3 and -3 are square roots of 9 because \( 3^2 = 9 \) and \( (-3)^2 = 9 \).
In our exercise, the number 0.04 has two square roots, which are +0.2 and -0.2. After squaring each, both results revert back to 0.04, verifying the roots:

• For the positive root \( +0.2^2 = 0.04 \).
• For the negative root \( (-0.2)^2 = 0.04 \).

This concept is vital not just in pure mathematics, but also in real-world applications where the direction or sign might have physical significance, such as in physics or economics.