Problem 44 - Intermediate Algebra

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt{(-7)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is 7.

Step 1: Identify the Expression Inside the Square Root

The expression inside the square root is \((-7)^2\). This indicates that we need to square the number \(-7\).

Step 2: Square the Number

Calculate \((-7)^2\). Squaring a number means multiplying it by itself: \((-7) \times (-7) = 49\).

Step 3: Take the Square Root

Next, find the square root of the resulting number. In this case, \(\sqrt{49}\).

Step 4: Calculate the Square Root

The square root of \(49\) is \(7\) since \(7 \times 7 = 49\). Square roots inherently return the positive root, considering all numbers are real.

Key Concepts

Exponents

Exponents are a way of representing repeated multiplication of the same number. When you see an expression like \((-7)^2\), the number \(-7\) is multiplied by itself. In the context of this exercise, the exponent "2" indicates two occurrences of multiplying \(-7\).

It is essential to follow the rule of multiplying negative numbers carefully:

• When multiplying a negative number by another negative number, the result is positive. Thus, \((-7) \times (-7) = 49\).
• This rule ensures that whenever you have an even exponent, the result will be positive, regardless of the sign of the base.

Exponents can apply to real numbers, integers, fractions, and even irrational numbers, demonstrating their wide applicability in mathematics.

In your calculations, always pay attention to the sign of the base number, especially when dealing with negative numbers and even exponents.

Real Numbers

Real numbers encompass the broadest category of numbers you will typically work with in basic mathematics. These include all rational and irrational numbers and cover:

• Positive numbers (e.g., \(3, 15.2\))
• Negative numbers (e.g., \(-7, -3.14\))
• Zero (\(0\))

The expression in the exercise, \((-7)^2\), involves a real number, as both integers and their resulting transformation through operations such as exponentiation (like being squared) result in real numbers.

Real numbers are crucial in calculus, geometry, and everyday measurements because they can represent almost anything you might encounter. They form the basis of the real number line, which allows you to visualize numbers in order from the largest to the smallest, including both positive and negative numbers with decimals and fractions.

Simplifying Expressions

Simplifying expressions involves transforming them into their simplest form while maintaining their value. In this exercise, you successfully simplify the expression \(\sqrt{(-7)^2}\) to \(7\), by following a few logical steps:

• First, calculate the exponent \((-7)^2\), resulting in \(49\).
• Next, take the square root of \(49\), which is \(7\).

The square root operation generally returns the principal (positive) root, especially for even-power expressions like squares, as real numbers can have both positive and negative square roots, yet by convention, we choose the non-negative root.

Simplifying expressions helps in performing mathematical operations more efficiently, ensuring that all variables and numbers are reduced to their cleanest form. This process is fundamental in solving algebraic equations and inequalities to gain insights into their behavior and solutions.