Problem 33 - Intermediate Algebra

Question

Evaluate each square root without using a calculator. See Objective 1 and Example 1. $$ \sqrt{121} $$

Step-by-Step Solution

Answer

The square root of 121 is 11.

Step 1: Understand the Problem

We need to find a number which, when multiplied by itself, equals 121.

Step 2: Apply Definition of Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. Thus, we need to find a number \( x \) such that \( x^2 = 121 \).

Step 3: Identify Familiar Square Numbers

Recall the perfect squares of small integers: \(1 \times 1 = 1\), \(2 \times 2 = 4\), ..., \(11 \times 11 = 121\). Notice that \(11^2 = 121\).

Step 4: Solution

Since \(11^2 = 121\), \( \sqrt{121} = 11 \). This is because 11 is the non-negative number which, when squared, results in 121.

Key Concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. This means, if you take an integer and multiply it by itself, the result is a perfect square. Understanding perfect squares is crucial because they allow us to easily calculate square roots, like in the exercise with \(\sqrt{121}\). For example, when you multiply the integer 11 by itself, you get 121, which is a perfect square. Recognizing these familiar numbers helps simplify many problems in mathematics.

Here are some examples of perfect squares:

• \(1^2 = 1\)
• \(2^2 = 4\)
• \(3^2 = 9\)
• \(4^2 = 16\)
• ... up to \(11^2 = 121\)

Knowing these, we can quick-check square roots without a calculator, as in the original exercise. This method is particularly helpful when working quickly or checking calculations.

Multiplication

Multiplication is a fundamental operation in mathematics that combines two numbers to get a product. In the context of finding square roots, understanding multiplication allows us to verify if a number is a perfect square. For instance, if you know how to multiply 11 by itself, you will see that the result is 121.

This operation can be expressed as:

• 11 \(\times\) 11 = 121

Verification through multiplication confirms that 121 is a perfect square because 11 is multiplied by itself to achieve this result. Recognizing when and how to apply multiplication in algebra will aid problem-solving and ensure accuracy in calculations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. When dealing with square roots, particularly, algebra helps to express relationships and solve for unknowns. For example, to find the square root of 121, we can set up the equation \(x^2 = 121\). This is an algebraic expression where \(x\) represents the unknown number we need to find.

To solve it, we look for a value of \(x\) that satisfies the equation. Familiarity with common expressions such as \(x^2\) assists in recognizing patterns and ensuring precise solutions to mathematical inquiries.