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Chapter 7: Problem 30

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{\frac{16}{49}}\)

Short Answer

Expert verified
\( \frac{4}{7} \)

Step by step solution

01

Identify the Square Root

Identify the square root operation to be simplified: \ \ \( \sqrt{\frac{16}{49}} \)
02

Separate the Numerator and the Denominator

Separate the square root of the fraction into the square root of the numerator and the square root of the denominator: \ \ \( \sqrt{\frac{16}{49}} = \frac{\sqrt{16}}{\sqrt{49}} \)
03

Simplify the Numerator

Simplify the square root of the numerator: \ \ \( \sqrt{16} = 4 \)
04

Simplify the Denominator

Simplify the square root of the denominator: \ \ \( \sqrt{49} = 7 \)
05

Combine the Results

Combine the simplified numerator and denominator: \ \ \( \frac{\sqrt{16}}{\sqrt{49}} = \frac{4}{7} \)

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Key Concepts

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

Square Root Properties
Understanding the properties of square roots is essential for simplifying them. A square root, denoted as \( \sqrt{...} \), represents a value that, when multiplied by itself, gives the original number. Let's delve into a few key properties:
  • The square root of a product: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)
  • The square root of a quotient: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) where \( b eq 0 \)
  • The square of a square root returns the original number: \( (\sqrt{a})^2 = a \)
These properties are helpful when simplifying expressions that involve square roots. For example, to simplify \( \sqrt{\frac{16}{49}} \), you can break it down into simpler parts according to these properties.
Simplifying Fractions
Simplifying fractions is a crucial step in making math problems more manageable. In general, a fraction consists of a numerator (the top part) and a denominator (the bottom part). When simplifying a fraction, the goal is to express it in its simplest form. To do this, you often follow these steps:
  • Factor both the numerator and the denominator into their prime factors.
  • Cancel out any common factors between the numerator and the denominator.
  • Recombine the simplified parts to get the final simplified fraction.
In the context of square roots, once the numerator and the denominator are simplified separately, you combine them back into a single fraction. For instance, simplifying \( \sqrt{16} \) gives 4 and simplifying \( \sqrt{49} \) gives 7. Recombining these results gives you the simplified fraction \( \frac{4}{7} \).
Numerator and Denominator
To master simplifying fractions, you need a solid understanding of numerators and denominators. These are the two main parts of a fraction:
  • Numerator: This is the top number in a fraction. It represents part of the whole.
  • Denominator: This is the bottom number in a fraction. It represents the total number of equal parts in the whole.
When dealing with square roots in fractions, you separate and simplify the numerator and denominator individually before recombining them. For example, in the fraction \( \frac{16}{49} \), 16 is the numerator and 49 is the denominator. By finding the square roots separately, \( \sqrt{16} \) simplifies to 4 and \( \sqrt{49} \) simplifies to 7, resulting in the simplified fraction \( \frac{4}{7} \).

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