Problem 12 evaluate each expression, or sta... [FREE SOLUTION]

Chapter 10: Problem 12

evaluate each expression, or state that the expression is not a real number. $$\sqrt{0.49}$$

Short Answer

Expert verified

The square root of 0.49 is 0.7.

Step by step solution

Check the nature of the number

The number given here is 0.49, which is a positive real number. Therefore, it is valid to take the square root of this number in real number system.

Calculate the square root

The square root of 0.49 can be calculated either using calculator or by recognizing that it is square of 0.7, hence its square root will be 0.7. So, $ \sqrt{0.49} = 0.7 $

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Key Concepts

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

Square Root

A square root is a value that, when multiplied by itself, gives the original number. For example, since 0.7 multiplied by 0.7 equals 0.49, the square root of 0.49 is 0.7.

This concept is represented by the radical symbol $ \sqrt{} $. The expression $ \sqrt{0.49} $ asks us to find a number that, times itself, equals 0.49.

Finding square roots is a common mathematical operation, especially when working with quadratic equations or geometric problems. Here's how you can explore square roots further:

Positive numbers usually have two square roots: one positive and one negative. For instance, both 0.7 and -0.7 squared equal 0.49. However, in most practical scenarios, the principal (positive) square root is used.
To calculate square roots, tools such as calculators or approximation methods like estimating between perfect squares can be useful.

Understanding how to compute square roots helps solve many algebraic problems effectively.

Positive Numbers

Positive numbers, as the name suggests, are numbers that are greater than zero. In the exercise, the number 0.49 is positive, which means it's not negative or zero. This characteristic influences whether certain operations can be performed on it.

Here are some simple ways to identify and work with positive numbers:

Positive numbers include all numbers on the number line that lie to the right of zero. This includes small fractions like 0.49, whole numbers like 1, 2, and very large numbers, like 1000 and beyond.
When calculating square roots or performing other arithmetic operations, knowing a number鈥檚 positivity helps determine the correct approach. Since square roots of positive numbers result in real numbers, $ \sqrt{0.49} $ is a real number.

Positive numbers are foundational in understanding basic arithmetic and algebra, and their properties are frequently used in problem-solving.

Real Number System

The real number system is a comprehensive set of numbers that includes both rational and irrational numbers. This system helps us understand and categorize every countable number. The square root expression $ \sqrt{0.49} $ is part of this system since 0.49 is a rational number.

The real number system consists of:

Rational numbers, which can be expressed as fractions of integers (e.g., 0.49 is $ \frac{49}{100} $).
Irrational numbers, which cannot be neatly expressed as fractions (like $ \pi $ or $ \sqrt{2} $).
Whole numbers, and integers, which are also subsets of the real number system.

Understanding how different types of numbers fit into the real number system can assist students in comprehending concepts like solving equations and measuring values.

By exploring the real number system, students can better grasp the broad nature of numbers and their relationships.

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91影视

evaluate each expression, or state that the expression is not a real number. $$\sqrt{0.49}$$

Short Answer

Step by step solution

Check the nature of the number

Calculate the square root

Key Concepts

Square Root

Positive Numbers

Real Number System

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