Problem 33 - Elementary and 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: Understanding Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. Hence, for \( \sqrt{121} \), we need to find a number \( x \) such that \( x \times x = 121 \).

Step 2: Identifying the Perfect Square

We recognize \( 121 \) as a perfect square. A perfect square is an integer that can be expressed as the square of another integer. We need to determine which integer squared equals 121.

Step 3: Checking the Integers

Start by checking small integers. \( 10 \times 10 = 100 \), which is less than 121. Next, check \( 11 \times 11 = 121 \). Here we see that 121 is indeed \( 11^2 \).

Step 4: Confirming the Root

Now that we have found \( 11 \), we confirm that \( \sqrt{121} = 11 \) because \( 11 \times 11 = 121 \). Therefore, the square root of 121 is 11.

Key Concepts

Perfect Squares

A perfect square is an easy concept to grasp once you understand it. It refers to numbers that can be obtained by multiplying an integer by itself. For example, when you multiply 3 by 3, you get 9. Hence, 9 is a perfect square. Similarly, 16 is a perfect square since it's the result of 4 multiplied by 4.

• 9 is 3 squared because 3 x 3 = 9
• 16 is 4 squared because 4 x 4 = 16
• 121 is 11 squared because 11 x 11 = 121

A perfect square will always end with a whole positive number when you take its square root. It's a nifty way to check if a number is a perfect square. Numbers like 1, 4, 9, 16, and 121 are just a few examples. Remember, perfect squares are always non-negative since squaring a number cannot result in a negative.

Evaluating Square Roots

Evaluating square roots is like reverse engineering a multiplication problem. When you evaluate a square root, you're looking for a number which, when squared, gives you the number under the square root sign. Consider the square root of 121. What number squared results in 121? Knowing your perfect squares can help solve these kinds of puzzles quickly.

• Step 1: Recognize that the aim is to discover a number that "squares" to the one given.
• Step 2: If 10 x 10 = 100, too low. Try moving up.
• Step 3: When 11 is squared; 11 x 11 equals 121.

Thus, the square root of 121 is 11 because that's the integer that squares back to 121. The process becomes straightforward when you regularly practice and familiarize yourself with perfect squares.

Integer Square Roots

An integer square root refers to a situation where taking the square root of a number results in a clean, whole number. This isn't always the case, but with perfect squares, it typically is. Some numbers yield beautiful, whole numbers when square-rooted. This makes recognizing perfect squares useful as you can quickly identify the integer they're associated with. For example, when you take the square root of 121 and get 11, that's an integer square root.

• An integer, like 11, is the result when the square root of 121 is evaluated.
• If you try non-perfect squares like 20, the result won't be a whole number.
• Working through these can be like solving a puzzle, trying different numbers until you find an exact match.

When evaluating square roots of perfect squares, you will always end up with integer values. Knowing this simplifies calculations and helps you identify patterns quickly.