Problem 7 - Algebra A Combined Function

Question

Find each square root. Assume that all variables represent nonnegative real numbers. $$ \sqrt{100} $$

Step-by-Step Solution

Answer

The square root of 100 is 10.

Step 1: Understand the Problem

The problem asks us to find the square root of 100, which means we need to determine a number that, when multiplied by itself, gives 100.

Step 2: Identify Perfect Squares

Recall that a perfect square is a number that is the square of an integer. Identify if 100 is a perfect square. 100 is a perfect square because it can be written as 10 multiplied by 10.

Step 3: Calculate the Square Root

Since 100 is the square of 10, the square root of 100 is simply the integer 10.

Key Concepts

Perfect Squares

A perfect square is a special kind of number which results from squaring an integer. When an integer is multiplied by itself, the result is termed a perfect square. For example, the number 4 is a perfect square because it is equal to 2 multiplied by 2, i.e., \(2^2 = 4\). This concept is quite important when finding square roots because it allows us to easily recognize which numbers have simple integer square roots.

Recognizing perfect squares can make calculating square roots much faster and simpler. If a number is a perfect square, its square root will always be an integer. Examples include numbers like 1, 4, 9, 16, 25, and so on. These numbers result from squaring integers like 1, 2, 3, 4, and 5, respectively.

Here are a few reasons understanding perfect squares is crucial:

• They simplify the process of finding square roots.
• They help in solving quadratic equations.
• They allow for easier calculations in areas involving geometry and algebra.

Integers

Integers are a broad category of numbers that include positive numbers, negative numbers, and zero. They are not fractions or decimals. Integers can be represented on the number line as whole steps either to the right or left of zero. Examples of integers are -3, 0, 4, and 7.

The importance of integers in the context of perfect squares and square roots is that a perfect square must always be derived from the square of an integer. Hence, when squaring an integer, the result is a positive integer (assuming the square itself is not 0). This highlights the crucial role integers play in arithmetic and algebra.

Understanding integers helps in:

• Performing basic arithmetic operations.
• Solving equations and inequalities.
• Modeling real-world problems that involve counting or linear computation.

Nonnegative Real Numbers

Nonnegative real numbers are numbers that are either positive or zero. They do not include negative numbers, though they may contain both integers and non-integers, such as fractions and irrationals like \( \sqrt{2} \). In mathematical expressions and calculations that involve square roots, we often assume all values represent nonnegative real numbers, especially as the square roots of negative numbers are not real.

Dealing with nonnegative real numbers ensures that solutions to square roots are valid and practical, as negative numbers do not naturally have real square roots. This is because no real number multiplied by itself will produce a negative result.

The importance of nonnegative real numbers is seen in:

• Ensuring meaningful solutions in mathematical problems like square roots.
• Avoiding complex numbers when unnecessary.
• Applying calculations to real-world quantifiable measurements, like length and area, which cannot be negative.