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Chapter 8: Problem 26

Simplify the given radical expression. $$ \sqrt{121} $$

Short Answer

Expert verified
\( \sqrt{121} = 11 \)

Step by step solution

01

Identify the Radical Expression

The expression we need to simplify is \( \sqrt{121} \). This is a square root, which implies we need to find a number, when multiplied by itself, equals 121.
02

Determine the Perfect Square

121 is a perfect square. To simplify \( \sqrt{121} \), we need to find the integer value whose square is 121. We check small numbers and find that \( 11 \times 11 = 121 \).
03

Simplify the Radical

Since 11 squared equals 121, the square root of 121 simplifies to 11. Therefore, \( \sqrt{121} = 11 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots
Square roots are fascinating mathematical tools. They let us find a number that, when multiplied by itself, reaches a specific value. For example, \( \sqrt{121} \) asks what number squared equals 121. Here's how to think about it:
  • Consider the operation "squaring" a number, which means multiplying it by itself.
  • Square rooting is the reverse 鈥� starting from the squared number, what was the original number?
  • In our case, the original number squared to get 121 is 11. So, the square root of 121 is 11.
Square roots are crucial for various mathematical calculations, providing a way to "unsquare" perfect squares and other numbers.
The symbol for a square root is \( \sqrt{} \), which is a handy representation for this operation. Understanding this helps to demystify numerous math problems quickly.
Perfect Squares
Perfect squares are numbers that you get when you multiply a whole number by itself. They play a key role when simplifying square roots. Let's break down what perfect squares are:
  • Start by taking any whole number, like 1, 2, or 3, and multiply it by itself. The results (1, 4, 9...) are perfect squares.
  • For instance, 11 is a whole number, and when squared (11 times 11), we get 121.
  • This means 121 is a perfect square because it derives from an integer squared.
Knowing about perfect squares is particularly useful for simplifying radicals. It allows you to spot when a number is a perfect candidate for easy radical simplification. If you see a number that's a perfect square, you instantly know its square root is going to be an integer. Recognizing these can turn complex math tasks into simpler ones.
Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, and more. Simplifying them is a way of making these expressions easier to work with. Here's a closer look at what they entail and how to simplify them:
  • Radicals often include expressions like \( \sqrt{a} \), where 'a' is a number or expression under the radical sign.
  • The goal of simplifying radicals is to "clean up" the expression by breaking it down into a simpler form. This often involves finding perfect squares.
  • In our example, \( \sqrt{121} \) is already a simple radical because 121 is a perfect square. The process becomes about finding the base number, which is 11 in this case.
Understanding radical expressions helps in a variety of fields in mathematics and science. Whether you're in algebra simplifying square roots or in calculus handling more complex root expressions, these basics form the building blocks of more advanced topics. Simplifying radicals can make equations less cluttered and much easier to solve or manipulate.

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Most popular questions from this chapter

Simplify the quotient, and write your answer in the form \(x^{r}\). $$ \frac{x^{-\frac{1}{2}}}{x^{-\frac{3}{5}}} $$

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