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Chapter 7: Problem 1

Simplify. Assume that variables represent positive real numbers. $$ \sqrt{100} $$

Short Answer

Expert verified
The square root of 100 is 10.

Step by step solution

01

Identify the Square Root

The given expression is \( \sqrt{100} \). This asks for the number which when multiplied by itself results in 100.
02

Calculate the Square Root

To find \( \sqrt{100} \), we need to determine which number multiplied by itself equals 100. We know that \( 10 \times 10 = 100 \), so the square root of 100 is 10.

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Key Concepts

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

Positive Real Numbers
Positive real numbers are numbers that are greater than zero and do not have any imaginary component. They are used in everyday math and are quite intuitive.
Here's what you need to know about positive real numbers:
  • Their value is always more than zero.
  • They can be whole numbers, fractions, or decimals.
  • They are often used in geometry, statistics, and many applications.
In problems involving square roots, like our original exercise, we assume all variables are positive real numbers unless stated otherwise. This is important because square roots of negative numbers lead us into the realm of imaginary numbers, something which we aren't addressing here. This assumption makes calculations straightforward and keeps us focused on the problem at hand. Remember, in these exercises, we're dealing purely with positive quantities.
Square Root
The square root represents a number that multiplies by itself to give a specific value. In our exercise, the square root of 100 is 10 because when you multiply 10 by itself, you get 100.
A square root is usually denoted by the symbol \( \sqrt{} \) and can be thought of in a few intuitive ways:
  • It is an inverse operation of squaring a number. For example, if 10 squared is 100, then the square root of 100 is 10.
  • Visually, it can be perceived as finding the side length of a square that has a certain area. So, a square with an area of 100 would have sides of length 10.
When simplifying square roots, especially when dealing with positive real numbers, the focus is on finding whole numbers that, when squared, match the value under the square root.
Multiplication
Multiplication is a fundamental part of understanding square roots. It involves adding a number to itself a certain number of times and is a cornerstone of many mathematical concepts, including finding square roots.
To illustrate, consider the expression \( 10 \times 10 = 100 \). Here, multiplication shows us that two tens equal 100.
Here's why it's essential in our exercise:
  • Square roots involve understanding which number, when multiplied by itself, equals the given number.
  • When multiplying two identical numbers, you find the square of that number. For instance, \( 10 \times 10 \) equals 100, helping us simplify \( \sqrt{100} \) to 10.
Thus, multiplication isn't just a calculation method鈥攊t's a critical part of understanding square roots. In simplifying square root problems, we oftentimes rely on our grasp of multiplication to deduce the answer.

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