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Chapter 8: Problem 5

Simplify. $$125--\sqrt{3}$$

Short Answer

Expert verified
The simplified expression is \(125 + \sqrt{3}\).

Step by step solution

01

Understand the Expression

The given expression is \(125 - -\sqrt{3}\). Notice that there is a double negative sign involved before the square root.
02

Apply Double Negative Rule

The rule of double negatives states that subtracting a negative is equivalent to addition. Therefore, \(125 - -\sqrt{3}\) becomes \(125 + \sqrt{3}\).
03

Write the Simplified Expression

Now, we simply write down the expression after resolving the double negative as \(125 + \sqrt{3}\). This cannot be simplified further as it contains an irrational number.

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Key Concepts

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

Double Negative Rule
Understanding the double negative rule is crucial when simplifying mathematical expressions. This rule simplifies the concept where two negative signs cancel each other out. When you see a double negative, as in expressions like
  • \(x - -y\)
think of it as a positive addition. This means you replace the two negative signs with a positive sign. Therefore,
  • \(x - -y = x + y\)
This can be remembered simply as a rule: "Subtracting a negative is the same as adding a positive." This rule is applicable in many algebraic expressions and is often used to simplify them.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol \(\sqrt{}\). For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9.
Square roots are involved in many areas of mathematics and are crucial for understanding quadratic equations, geometry, and more.
Whenever you encounter a square root in an expression, like \(\sqrt{3}\), it is implying a number which when squared, results in 3. Simplification involving square roots might not always lead to a whole number.
  • This makes such numbers special and often left in their simplest form, to give precise results.
Irrational Numbers
Irrational numbers are numbers that cannot be expressed exactly as a simple fraction. These numbers have non-repeating, non-terminating decimal forms. A classic example of an irrational number is \(\pi\), but square roots of numbers that are not perfect squares, like \(\sqrt{3}\), also fall into this category.
Irrational numbers are important because they help us describe the real world with a higher degree of accuracy, especially in measurements that are not exact whole numbers.
  • In the simplified expression \(125 + \sqrt{3}\), \(\sqrt{3}\) remains irrational.
  • This means it cannot be broken down further into a simpler form or a fraction.
Understanding irrational numbers allows for a complete grasp of more complex mathematical operations and analyses.

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