Problem 34 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{9}{49}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( \sqrt{\frac{9}{49}} \) is \( \frac{3}{7} \).

Step 1: Simplify Numerator

The numerator under the square root is a perfect square. The square root of 9 is \( \pm 3 \). However, in this context, we consider the positive root as it's a length measurement.

Step 2: Simplify Denominator

The denominator under the square root is also a perfect square. The square root of 49 is \( \pm 7 \). In this context, we will consider the positive root as it's a length measurement.

Step 3: Simplify the Expression

Thus the square root of the fraction \( \frac{9}{49} \) is \( \frac{\sqrt{9}}{\sqrt{49}} \), which is \( \frac{3}{7} \).

Key Concepts

Simplifying Radical Expressions

Understanding how to simplify radical expressions is essential to mastering algebra and higher-level math courses. A radical expression involves roots, such as square roots or cube roots. Simplifying these expressions makes them easier to work with and can involve a few different steps depending on what is under the radical.

To start simplifying a radical expression, like the square root of a fraction, you should simplify the numerator (top number) and the denominator (bottom number) separately. If they are perfect squares (more on perfect squares below), you take the square root of each part. Remember, when you take the square root of a number, you're looking for the number that, when multiplied by itself, gives back the original number.

In our example, \( \sqrt{\frac{9}{49}} \) consists of the perfect squares 9 and 49. By breaking it down, we get \( \frac{\sqrt{9}}{\sqrt{49}} \) which simplifies to \( \frac{3}{7} \) since 3 is the square root of 9 and 7 is the square root of 49. It is crucial to only take the positive roots since we're dealing with lengths, and in most contexts, negative length doesn't make sense. Simplifying radical expressions often involves recognizing these perfect squares and understanding how to handle square roots.

Perfect Squares

Perfect squares play a pivotal role in mathematics. They are integers that are the square of an integer. For example, 1, 4, 9, 16, and so on, are perfect squares since they can be written as 1^2, 2^2, 3^2, 4^2, respectively. Knowing the perfect squares up to at least 12^2 can significantly speed up the process of simplifying radical expressions.

In the example we're exploring, 9 and 49 are both perfect squares. They are, respectively, the squares of 3 and 7. When you encounter a radical expression that contains perfect squares, it simplifies the process because you can directly take their square roots without additional steps. It's like unraveling a neatly wrapped package - once you identify the perfect square, you can easily find its root.

Quick Tips for Working with Perfect Squares:

• Memorize the list of perfect squares from 1 to at least 12 for quick recall.
• Use prime factorization for larger numbers to identify if they're perfect squares.
• Remember that the square root of a perfect square is always an integer.

Understanding perfect squares is not only vital for simplifying radicals but also for other topics in mathematics, such as quadratic equations and properties of numbers.

Rationalizing Denominators

When working with fractions that contain radicals, especially in the denominator, you may often be required to 'rationalize' the denominator. Rationalizing a denominator means to eliminate radicals from it, achieving a simpler or more 'rational' form. To be clear, the term 'rational' in mathematics refers to numbers that can be expressed as the ratio of two integers.

To rationalize a denominator with a square root, you multiply the numerator and denominator by the same square root that is in the denominator. This process will remove the square root from the denominator without changing the value of the original expression. This step is important because it often makes further arithmetic easier and is the preferred way of presenting solutions.

In our example, we do not need to rationalize because after simplifying the square root of \( \frac{9}{49} \) we are left with the fraction \( \frac{3}{7} \) which has no radicals in the denominator. However, if we had started with a denominator like \( \sqrt{5} \) we would multiply the fraction by \( \frac{\sqrt{5}}{\sqrt{5}} \) to rationalize it. Rationalizing denominators is an important skill and is widely used in algebra and calculus for simplifying expressions and solving equations.