Problem 50 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified form is 7.

Step 1: Understand the Square inside the Square Root

The expression \((-7)^2\) means that we are squaring the number \(-7\). Squaring a number means multiplying the number by itself: \((-7)\times(-7)\).

Step 2: Calculate the Square of the Negative Number

When multiplying \(-7\) by itself, we get a positive result, because multiplying two negative numbers results in a positive number: \[(-7) \times (-7) = 49\].

Step 3: Simplifying the Square Root

Now, take the square root of 49. The square root of a number is a value that, when multiplied by itself, gives the original number. Thus, \( \sqrt{49} = 7 \).

Step 4: Final Step: Conclude the Simplification

Since we have reduced the expression \( \sqrt{(-7)^2} \) to \( \sqrt{49} = 7 \), the simplified form of the original expression is \(7\).

Key Concepts

Understanding the Square Root

The square root is a mathematical function that helps us find a number which, when multiplied by itself, returns the original number inside the root. For example, in the expression \(\sqrt{49}\), we are looking for a value that when squared gives us 49. The answer is 7 because \(7 \times 7 = 49\).

• The square root is denoted by the symbol \(\sqrt{}\).
• It is applicable to both perfect squares and non-perfect square numbers.
• Square roots can be real or complex numbers, but in most basic mathematics, they are presented as real numbers.

Understanding square roots is essential in simplifying expressions like \(\sqrt{(-7)^2}\), where you deal with negatives correctly.

Mastering Squaring Numbers

Squaring a number means multiplying the number by itself. It is written as \(n^2\), where \(n\) is the base. If we square the number \(-7\), it looks like this: \((-7)^2\), meaning \((-7) \times (-7)\).

• Squaring any number always results in a non-negative value, either positive or zero.
• The operation neutralizes any negative sign because multiplying two negatives equals a positive.

For example, squaring \(-7\), we multiply:

• First step: \(-7 \times -7\)
• Second step: Result is 49

Remember, squaring is essential before taking the square root in expressions like \(\sqrt{(-7)^2}\).

Deciphering Negative Numbers

Negative numbers are numbers less than zero and often represented with a minus (-) symbol. In mathematics, they have a critical role, especially in squaring and taking roots.

• When two negative numbers are multiplied, the result is always positive.
• In expressions, negative roots are not considered in basic real number simplification.

For instance, \((-7) \times (-7) = 49\). Here, even though \(-7\) is negative, the result of its square is positive because of multiplication rules.

The World of Real Numbers

Real numbers include all the numbers that can be found on the number line. They encompass both rational and irrational numbers, including fractions and whole numbers. In mathematics, simplifying expressions typically assumes all variables represent real numbers. This means excluding imaginary or complex numbers in basic calculations like \(\sqrt{(-7)^2}\).

• Real numbers can be positive, negative, or zero.
• They serve as the foundational block for further advanced mathematical studies.
• Simplifying an expression typically means reducing it under the assumption that the numbers involved are real.

Being aware of real numbers allows us to understand results like arriving at 7 from \(\sqrt{49}\) without confusion over imaginary components.