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Chapter 7: Problem 32

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{\frac{13}{49}}\)

Short Answer

Expert verified
\( \frac{\sqrt{13}}{7} \)

Step by step solution

01

Break Down the Radicand

Identify the numerator and the denominator inside the square root. The given expression is \(\frac{13}{49}\), so the numerator is 13 and the denominator is 49.
02

Simplify the Square Root of the Numerator

Since 13 is a prime number, it does not have any square factors other than 1. Thus, \[ \text{Simplify} \sqrt{13} = \sqrt{13} \]
03

Simplify the Square Root of the Denominator

49 is a perfect square because \[ 49 = 7^2 \] Hence, the square root of 49 is 7: \[ \sqrt{49} = 7 \]
04

Combine the Simplified Parts

Using the results from Steps 2 and 3, combine the simplified parts under a single square root: \[ \sqrt{\frac{13}{49}} = \frac{\sqrt{13}}{7} \]

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Key Concepts

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

Understanding the Square Root
A square root is a special number that, when multiplied by itself, gives back the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\). The symbol for the square root is \(\sqrt{}\). It's important because it helps simplify complex expressions and find shorter ways to represent large numbers. In the given exercise, we鈥檙e working with the square root of a fraction, which might seem tough but can be broken down into simpler steps.
Radicand - The Number Inside the Square Root
The number or expression inside the square root symbol is called the radicand. In our example, the radicand is \(\frac{13}{49}\). When we look at the radicand, we see the fraction that needs to be simplified. We break it down into two parts - the numerator (13) and the denominator (49). Simplifying each part separately helps us manage the calculation more effectively.
Prime Numbers and Their Role
Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. In simpler terms, they cannot be divided evenly by any number other than 1 and the number itself. For example, 2, 3, and 5 are prime numbers. In our solving process, we find that 13 is a prime number. This indicates it does not have any square factors (like 4 or 9). So, the square root of 13 remains \(\sqrt{13}\), because it cannot be simplified further.
Perfect Squares 鈥� Making Simplification Easier
A perfect square is a number that is the square of an integer. This means it can be written as some integer multiplied by itself. For example, 9 is a perfect square since \(3 \times 3 = 9\). In our problem, the denominator 49 is a perfect square because \(7 \times 7 = 49\). This allows us to simplify \(\sqrt{49}\) to 7 easily. Using perfect squares wherever possible can make the simplification process faster and easier.

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