Problem 16 - Prealgebra

Question

List all square roots of the given number. If the number has no square roots, write “none”. 400

Step-by-Step Solution

Answer

The square roots of 400 are 20 and -20.

Step 1: Identify the context

We need to find all numbers whose square is 400. The square root of a number \( x \) is a value that, when multiplied by itself, results in \( x \).

Step 2: Calculate the positive square root

To find the positive square root of 400, consider a number whose square is 400. Calculate \( \sqrt{400} = \sqrt{20^2} = 20 \).

Step 3: Consider negative square roots

For every positive square root, there is a corresponding negative square root, because \((-20)^2 = 400\). Thus, the negative square root of 400 is -20.

Step 4: List all square roots

The number 400 has two square roots: the positive square root, 20, and the negative square root, -20.

Key Concepts

Positive Square Root

Finding the positive square root of a number is like asking, "What number multiplied by itself gives the original number?" For instance, in the case of 400, we are looking for a number that when squared (i.e., multiplied by itself) equals 400.

To find this, you use the square root function, typically denoted as \( \sqrt{} \) in mathematics. The positive square root of 400 is determined by finding the number that squares to 400 without any regards to sign. It's given as \( \sqrt{400} \). This equals 20 since \( 20 \times 20 = 400 \).

Remember:

• The positive square root is always non-negative.
• When you see the square root symbol without a sign, it refers to the positive square root by default.

Negative Square Root

Every positive square root has a negative counterpart because squaring involves both positive and negative numbers. Consider this: if multiplying 20 by itself gives 400, doesn’t multiplying -20 by itself do the same?

The negative square root of 400 is simply the negative value of its positive square root — that's \(-20 \). Square -20, and you still get 400, since \((-20) \times (-20) = 400\).

Things to remember about negative square roots:

• They are represented with a negative sign in front of the square root symbol.
• Negative square roots have equal magnitude but opposite sign to their positive square root partners.

Never confuse the two; even though they both square to the same result, their positions on the number line are very different, making them distinct solutions.

Perfect Square

A perfect square is a number that is the square of an integer. In simple words, if you can find an integer such that multiplying it by itself gives your number, then your number is a perfect square.

Taking 400 as an example, we find that both 20 and -20 are perfect squares of different integers (20 and -20), because:

\[ 20 \times 20 = 400 \]
\[ (-20) \times (-20) = 400 \]

This means 400 is a perfect square.

Key points about perfect squares:

• They result from multiplying an integer by itself.
• Every perfect square has two square roots: one positive and one negative.

Understanding this concept helps in recognizing perfect squares quickly and efficiently.