Problem 29 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$0.09$$

Step-by-Step Solution

Answer

The square roots of 0.09 are \(+0.3\) and \(-0.3\).

Step 1: Determine the square roots

A square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). In this case, both \(+0.3\) and \(-0.3\) are square roots of \(0.09\), since \((+0.3)^2\) equals \(0.09\) and \((-0.3)^2\) also equals \(0.09\).

Step 2: Verify the square roots

Check if the obtained values are indeed the roots by squaring each root. Squaring \(+0.3\), we get \(+0.3 * +0.3 = 0.09\), and squaring \(-0.3\), we get \(-0.3 * -0.3 = 0.09\). Therefore, both \(+0.3\) and \(-0.3\) are the square roots of \(0.09\).

Key Concepts

Square Numbers

Square numbers are special types of numbers obtained by multiplying a number by itself. They are also known as perfect squares. For example, when you take the number 3 and multiply it by itself, the result is 9, which is a square number. In another instance, multiplying 0.3 by 0.3 gives us 0.09, which is also a square number.
Square numbers are important because they form the basis for understanding square roots and are used in many mathematical applications. Recognizing square numbers can help one determine whether a given number has exact square roots or not.

Verification of Solutions

After calculating square roots, it is always a good idea to verify the answers to ensure accuracy. Verification involves checking the results by performing specific calculations. In the context of square roots, this means squaring the determined roots to see if you get back the original number.
For example, with the number 0.09, if we propose the roots to be +0.3 and -0.3, we can verify these claims by squaring each. When squaring +0.3, we compute \((+0.3) \times (+0.3) = 0.09\). Doing the same with -0.3, \((-0.3) \times (-0.3) = 0.09\). Both calculations confirm that 0.09 is indeed the original number, verifying that +0.3 and -0.3 are correct square roots.
Verification provides an extra layer of assurance in mathematical problems and reinforces the reliability of the computed solutions.

Negative and Positive Roots

Square roots naturally come in pairs: one positive and one negative. This is because squaring a positive or a negative number always results in a positive product. It’s essential to consider both roots to have a complete answer.
When we talk about the square root of 0.09, both +0.3 and -0.3 qualify. Even though they are opposite in sign, squaring either of them results in 0.09.
This duality of roots is a fundamental property of numbers and is vital for solving equations, understanding graphing, and more advanced mathematics. It also highlights how mathematics can express symmetry and balance, as every positive root has a negative twin.