Problem 64 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: Write the Radicals Separately

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

Step 2: 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 \( \sqrt{9} = 3 \).

Step 3: Write Them Together

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

Key Concepts

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.