Problem 42 - Elementary Algebra

Question

For the following problems, find the two square roots of the given number. $$ 0.16 $$

Step-by-Step Solution

Answer

Answer: The two square roots of 0.16 are 0.4 and -0.4.

Step 1: Determine the positive square root

To find the positive square root of 0.16, we can use the following equation: $$ x^2 = 0.16 $$ Taking the square root of both sides, we get: $$ x = \sqrt{0.16} $$ Calculating, we get: $$ x = 0.4 $$

Step 2: Determine the negative square root

To get the negative square root of 0.16, simply put a negative sign in front of the positive square root we just found: $$ -x = -0.4 $$

Step 3: Present the final answers

The two square roots of 0.16 are 0.4 and -0.4.

Key Concepts

Positive Square Root

Every positive real number has a unique positive square root. This means that if you have a number like 0.16, it is the result of multiplying some other number by itself. The positive square root is the non-negative number that, when squared, gives the original number back.

For instance, the positive square root of 0.16 is found using the equation \( x^2 = 0.16 \). When we solve for \( x \), we get \( x = 0.4 \), since \( 0.4 \times 0.4 = 0.16 \). This process of finding a square root is essentially asking what number multiplied by itself results in the number we started with.

• When dealing with square roots, you typically denote the positive square root using the square root symbol \( \sqrt{} \).
• The positive square root is always greater than or equal to zero.

By finding this value, you discover the positive side of a number's square root, helping to define its place on the number line.

Negative Square Root

Just like positive square roots, numbers also have negative square roots. You might wonder how this can be since when you multiply two negative numbers, the result is positive.

For a number like 0.16, its negative square root would be \( -0.4 \). This is because \( (-0.4) \times (-0.4) = 0.16 \) as well. To find the negative square root, you simply take the positive square root and add a negative sign in front of it.

• Negative square roots are less than zero.
• They offer a complete picture of a number's square roots by accounting for the symmetric nature of squares on the number line.

It's important for students to recognize both the positive and negative roots, as they represent critical values in many mathematical problems, particularly those involving quadratic equations.

Mathematics Education

Understanding square roots is a key part of mathematics education. It's not just about performing calculations but building a solid foundation in number sense and algebraic thinking.

Teaching square roots involves more than memorizing processes; it includes encouraging students to visualize and reason about numbers in various ways.

• Engage students with practical examples, like finding the side length of a square when given the area.
• Introduce tools such as number lines to help visualize where both the positive and negative square roots fall.
• Discuss real-world applications where square roots are used, such as in engineering and computer sciences.

Mathematics education should empower students to understand fundamental concepts thoroughly, so they can apply them confidently in more complex situations. Encouraging experimentation and questioning can lead to deeper understanding and retention of concepts like square roots.