Problem 1 - Prealgebra

Question

In Exercises 1-16, list all square roots of the given number. If the number has no square roots, write “none”. 256

Step-by-Step Solution

Answer

The square roots of 256 are 16 and -16.

Step 1: Understand the Concept of Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. A number can have both positive and negative square roots.

Step 2: Recall the Calculation of Square Roots

For any number \( x \), the square roots are the solutions to the equation \( x^2 = a \). When \( a \) is a perfect square, \( x \) will have a whole number and its opposite as solutions.

Step 3: Identify if 256 is a Perfect Square

Identify if 256 can be expressed as the square of an integer. In this case, if a whole number multiplied by itself results in 256.

Step 4: Calculate the Square Root of 256

Find out what number squared equals 256. Test numbers sequentially: since \(16^2 = 256\), then 16 is a square root.

Step 5: Account for Negative Square Root

Recall that both positive and negative versions of a number can be square roots. If \( 16^2 = 256 \), then \((-16)^2 = 256\) as well.

Step 6: List Both Square Roots

Since both positive and negative 16 satisfy the equation, the square roots of 256 are 16 and -16.

Key Concepts

Perfect Square

A perfect square is a number that can be expressed as the product of an integer with itself. For example, 256 is a perfect square because it equals 16 times 16.
A number is considered a perfect square if you can find an integer "n" such that \( n^2 = \text{the number} \).

• Examples of perfect squares include 1, 4, 9, 16, 25, 36, etc.
• These numbers result from multiplying an integer by itself: \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), and so on.

Identifying perfect squares is essential because they always have integer square roots.

Calculation of Square Roots

Calculating a square root involves determining which number, when multiplied by itself, equals the original number.
When the original number is a perfect square, its square root will be a whole number.

• The square root of 256 is 16 because when you multiply 16 by itself (\( 16 \times 16 \)), the result is 256.
• Calculations for non-perfect squares are more complex and may result in decimal values.

Understanding this process is crucial, as the ability to find square roots is a fundamental skill in mathematics.

Positive Square Root

The positive square root of a perfect square is simply the principal square root of that number. It is typically the non-negative result.
For example, for the number 256, the positive square root is 16 because \( 16^2 = 256 \).
Every positive number greater than zero has a positive square root.

• The symbol used for square root is "\( \sqrt{} \)", so "\( \sqrt{256} \)" equals 16.
• Positive square roots are useful in various calculations where non-negative values are needed.

It's essential to understand that the positive square root is the customary result in most mathematical contexts unless stated otherwise.

Negative Square Root

Just as numbers can have positive square roots, they can also have negative square roots. The negative square root of a number is simply the opposite of the positive one.
For the number 256, the negative square root is -16 because \( (-16)^2 = 256 \).
Every positive number has a corresponding negative square root.

• When you see an expression like "\( x^2 = 256 \)", both \( x = 16 \) and \( x = -16 \) are solutions.
• It's important to remember that squaring any real number – whether positive or negative – results in a non-negative number.

The negative square root is as valid as the positive, particularly in equations that require listing all possible roots.