Problem 30 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$0.16$$

Step-by-Step Solution

Answer

The square roots of 0.16 are \( +0.4 \) and \( -0.4 \), which are verified as a square of each root gives 0.16.

Step 1: Find the Square Root

To determine the square root of a number, use the square root function. The square root of 0.16 is either \( +0.4 \) or \( -0.4 \) because a number can have two square roots: one positive and one negative. Thus, the two roots of 0.16 are \( +0.4 \) and \( -0.4 \).

Step 2: Verify the Roots

Verify the roots by squaring each of them. The square of \( +0.4 \) is 0.16 and the square of \( -0.4 \) is also 0.16.

Key Concepts

Positive Square Root

When we speak of the positive square root, we're referring to the non-negative number that, when multiplied by itself, gives the original number. For the number 0.16, calculating the positive square root involves finding a number that when squared equals 0.16.

In mathematical terms, this is noted as \(\sqrt{0.16}\ = +0.4\). This means 0.4 times 0.4 gives us back 0.16. The positive square root is often what people mean when they casually refer to "the square root" of a number. It is symbolized by the radical sign (√) without a negative sign, indicating the non-negative result. Calculators typically offer this result when you compute the square root function.

Understanding the positive square root is crucial because in many real-world applications, like calculating area, distance, or financial metrics, we are often only interested in the non-negative result.

Negative Square Root

The negative square root is just as important as its positive counterpart. It refers to the negative version of the number that also satisfies the equation of being squared to equal the original number. For example, the negative square root of 0.16 is written as \(-\sqrt{0.16}\ = -0.4\).

Although negative square roots aren't often used as frequently as positive ones in everyday scenarios, they are just as valid in mathematics. This reflects the fact that both \(+0.4\) and \(-0.4\) are solutions to the equation \(x^2 = 0.16\).

It's essential to remember that whenever dealing with square roots, a number may have two roots: a positive and a negative one. These are sometimes referred to as the principal and secondary roots. Incorporating both helps in comprehensive problem-solving, especially in algebra and calculus contexts.

Verification of Roots

Verification of roots is the process of confirming that the calculated square roots are indeed correct. This process is straightforward: square each of the roots and check if you arrive back at the original number.

For our number 0.16, this involves checking both roots, \(0.4\) and \(-0.4\):

• Square \(0.4\): \((0.4)^2 = 0.16\)
• Square \(-0.4\): \((-0.4)^2 = 0.16\)

Both calculations confirm that our roots are accurate because they both return the original number, 0.16. Verification is an essential step, especially in academic settings, because it ensures that errors in computation have not been made.

To sum up, always square both the positive and negative roots when verifying, as both contribute toward understanding the entire solution in mathematical problems.