Problem 17 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \pm \sqrt{0.64} $$

Step-by-Step Solution

Answer

The expression evaluates to \( +0.8 \) and \( -0.8 \).

Step 1: Understand the Expression

The expression given is \( \pm \sqrt{0.64} \), which states that we need to find both the positive and negative square roots of 0.64.

Step 2: Compute the Square Root

Calculate \( \sqrt{0.64} \). Recall that 0.64 is a perfect square since it can be rewritten as \( (0.8)^2 \). Thus, \( \sqrt{0.64} = 0.8 \).

Step 3: Apply the Plus-Minus Sign

The expression \( \pm \sqrt{0.64} \) indicates that we should consider both the positive and negative results of the square root calculation. Therefore, the results are \( +0.8 \) and \( -0.8 \).

Step 4: Write the Final Answer

The evaluation of \( \pm \sqrt{0.64} \) results in two numbers, \( +0.8 \) and \( -0.8 \).

Key Concepts

Real Numbers

Real numbers include all the numbers we typically use in everyday life. They cover a wide range of possibilities, including:

• Whole numbers, like 0, 1, 2, etc.
• Rational numbers, such as fractions like \( \frac{1}{2} \); they can be expressed as a fraction of two integers.
• Irrational numbers, such as \( \pi \) or \( \sqrt{2} \), which cannot be represented as a simple fraction.
• Decimals, like 0.5 or -2.64.

Real numbers also include negative numbers, which are simply: numbers less than zero. They can represent values like -1 or -3.75. In the context of square roots, real numbers are essential because they provide potential values for both positive and negative results. For example, even if we extract a square root from a positive number, we may get results that can be positive or negative—just like in the problem \( \pm \sqrt{0.64} \)—where both 0.8 and -0.8 are valid real numbers.
Real numbers allow for a full spectrum of possibilities, making them crucial for comprehending solutions to square root problems where we consider both polarities.

Positive and Negative Roots

When we see an expression with a square root, such as \( \pm \sqrt{x} \), it means we are considering both the positive and negative square roots of \ 'x\'.
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 \( 3 \times 3 = 9 \). However, there is also a negative counterpart: -3, because \( (-3) \times (-3) = 9 \) as well.This is why when calculating square roots in mathematics, it is important to account for both the positive and negative roots.

• Positive square root: Written as \( \sqrt{x} \), gives a non-negative number.
• Negative square root: Expressed with a negative sign in front, as \( -\sqrt{x} \), gives a negative number.

In the exercise, \( \pm \sqrt{0.64} \) implies both +0.8 and -0.8 are valid solutions because 0.64 is a positive number, hence allowing for both root values.

Perfect Square

A perfect square is a number that is the square of an integer or, in some cases, a simple decimal. This means a perfect square can be the area of a square whose side length is an exact number, either whole or decimal.
For example:

• 4 is a perfect square because it is \( 2^2 \).
• 16 is a perfect square since \( 4^2 = 16 \).
• 0.64 is a perfect square because it equals \( (0.8)^2 \).

Perfect squares are significant because when taking a square root of such numbers, we can easily find the square root because it corresponds directly to the number squared to make it.
The benefit is that identifying a perfect square simplifies the process of finding a square root. For instance, knowing that 0.64 is \( (0.8)^2 \) swiftly provides us with the knowledge that \( \sqrt{0.64} = 0.8 \). It turns out quite handy in algebra because it allows for quick simplification and verification in equations and problems involving square numbers.