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Chapter 11: Problem 64

Simplify the radical expression. $$\sqrt{\frac{11}{9}}$$

Short Answer

Expert verified
The simplified form of \( \sqrt{\frac{11}{9}} \) is \( \frac{\sqrt{11}}{3} \).

Step by step solution

01

Write the Radicals Separately

We can separate the fraction under the square root into two separate roots. The expression becomes \( \sqrt{11/9} = \sqrt{11} / \sqrt{9} \).
02

Simplify the Separate Radicals

The square root of 11 remains as it is because 11 is a prime number which doesn't have any perfect square factors. We will denote this as \( \sqrt{11} \). On the other hand, 9 is a perfect square, so its square root is a whole number. Applied square root for 9, which is \( \sqrt{9} = 3 \).
03

Write Them Together

Combining both results, we can get the final simplified form. So, we can write \( \sqrt{11/9} = \sqrt{11} / \sqrt{9} = \sqrt{11}/3 \).

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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 fundamental concepts in mathematics. A square root of a number is a value that, when multiplied by itself, gives that number. For example, the square root of 4 is 2, because 2 multiplied by 2 equals 4. Square roots are represented by the radical symbol \(\sqrt{}\). If you encounter \(\sqrt{9}\), it asks you what number multiplied by itself gives 9, which is 3.
  • They can be applied to whole numbers, fractions, or even decimals.
  • If the number under the square root (known as the radicand) is a perfect square, the square root is an integer.
  • If not, the square root remains in its radical form or can be approximated as a decimal.
Square roots can also be separated when dealing with fractions. This allows us to simplify expressions involving fractions, just as we did in our example with \(\sqrt{\frac{11}{9}} = \sqrt{11}/\sqrt{9}\). This can make solving problems much easier.
Prime Numbers
Prime numbers are integers greater than 1, and they have no divisors other than 1 and themselves. In other words, a prime number can be divided, without a remainder, only by 1 and the number itself. The number 11, for instance, is prime because it cannot be divided evenly by any number other than 1 and 11.
  • Prime numbers are the building blocks of all natural numbers because any number can be expressed as a product of prime numbers.
  • Only positive whole numbers can be prime; negative numbers and non-integers cannot be.
This understanding is crucial in simplifying radicals, especially when examining the radicand for factors. If a number is prime, like 11, it cannot be broken down further or simplified. That's why \(\sqrt{11}\) remains as it is.
Perfect Squares
Perfect squares are numbers that result from squaring an integer. That means a perfect square is a number that can be expressed as the product of an integer with itself. For example, 9 is a perfect square because it is the product of 3 times 3.
  • Common perfect squares include 1, 4, 9, 16, 25, and so on.
  • Perfect squares are always non-negative since the square of any real number is positive.
Recognizing perfect squares is key in simplifying square roots. When you see a number under a square root that is a perfect square, like 9 under \(\sqrt{9}\), you can simplify it to its integer root, which is 3 in this case. This makes calculations more straightforward and often leads to exact answers.

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Most popular questions from this chapter

Evaluate the expression. $$\left(3^{3}\right)^{2}$$

You will write and simplify a general expression for the average speed traveled when making a round trip. Let \(d\) represent the one-way distance. Let \(x\) represent the speed while traveling there and let \(y\) represent the speed while traveling back. Write an expression for the total time for the round trip. Use addition to write your answer as a single rational expression.

Use the following information. You are choosing a business partner for a student lawn-care business you are starting. It takes you an average of 35 minutes to mow a lawn, so your rate is 1 lawn in 35 minutes or \(\frac{1}{35}\) of a lawn per minute. Let \(x\) represent the average time (in minutes) it takes a possible partner to mow a lawn. Write an expression for the partner's rate (that is, the part of a lawn the partner can mow in 1 minute). Then write an expression for the combined rate of you and your partner (the part of a lawn that you both can mow in 1 minute if you work together).

You will look for a pattern. What happens to the values of \(\frac{x^{2}+6}{x+2},(x-2),\) and \(\frac{10}{x+2}\) as \(x\) increases?

Simplify the expression. (Review \(8.3 \text { for } 11.7)\) $$\frac{4 m^{2}}{6 m}$$
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