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Chapter 7: Problem 23

Simplify. $$ -\sqrt{\frac{16}{81}} $$

Short Answer

Expert verified
-\( -\frac{4}{9} \)

Step by step solution

01

Identify the expression under the square root

The term under the square root is \( \frac{16}{81} \). This needs to be simplified first.
02

Simplify the fraction

Simplify both the numerator and the denominator separately. The square root of 16 is 4, and the square root of 81 is 9. Thus, \( \frac{\text{鈭殅16}{\text{鈭殅81} = \frac{4}{9} \).
03

Apply the negative sign

Include the negative sign in front of the simplified fraction obtained from the previous step to get \( -\frac{4}{9} \).
04

Final expression

The simplified form of \( -\text{鈭殅\left(\frac{16}{81}\right) \) is \( -\frac{4}{9} \).

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Key Concepts

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

rational expressions
A rational expression is like a fraction. It has a numerator (the top part) and a denominator (the bottom part). These parts can be numbers, variables, or more complex mathematical expressions. In many math problems, you need to simplify these rational expressions to make them easier to work with.
  • For instance, in our problem, the rational expression is \( \frac{16}{81} \).
  • By simplifying, you make problems clearer and easier to solve.
Remember, the focus when working with rational expressions is to identify and reduce both the numerator and the denominator effectively.
numerator and denominator
The numerator and the denominator are crucial parts of fractions and rational expressions. Understanding these can help you simplify and solve more complex math problems.
  • The numerator is the top part of the fraction (here, it's 16).
  • The denominator is the bottom part of the fraction (in our case, it's 81).
  • When simplifying, you often break down these parts separately before combining them.
For example, the square root of the numerator 16 is 4, and the square root of the denominator 81 is 9. Putting these together gives us \( \frac{4}{9} \). So, knowing how to handle the numerator and denominator can simplify what looks like a complex fraction into something easy to manage.
negative sign
Negative signs in math can sometimes be tricky, but they follow simple rules. When you see a negative sign in front of a square root or a fraction, it means the entire value is negative.
  • In our specific problem, the negative sign in front of \( \sqrt{\frac{16}{81}} \) means we need to apply it to our simplified fraction.
  • So, instead of just \( \frac{4}{9} \), we have \( - \frac{4}{9} \).
This negative sign doesn't affect the process of simplification but must be included in the final answer. By keeping track of the negative signs, you ensure the correctness and clarity of your math solutions.

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