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Chapter 8: Problem 86

Simplify. $$ 81--\sqrt{3} $$

Short Answer

Expert verified
The simplified expression is \(81 + \sqrt{3}\).

Step by step solution

01

Simplify Double Negative

The expression given is \(81--\sqrt{3}\). The first step is to simplify the double negative, which means "minus a negative," is equivalent to addition. Therefore, \(81--\sqrt{3}\) becomes \(81 + \sqrt{3}\).
02

Verify Simplification

Now, let's check our simplified expression from the previous step to ensure that no further simplification is possible. The expression \(81 + \sqrt{3}\) cannot be simplified any further, as \(81\) is a whole number and \(\sqrt{3}\) is an irrational number. The final simplified form will remain \(81 + \sqrt{3}\).

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Key Concepts

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

Double Negative
Whenever you see a double negative in mathematics, like in the expression \(81--\sqrt{3}\), it's a key opportunity to simplify. A double negative means adding instead of subtracting. Think of it as having two minus signs that cancel each other out.
  • For example, \(-(-x)\) simply becomes \(+x\).
  • In this exercise, \(81--\sqrt{3}\) becomes \(81 + \sqrt{3}\).
This simplification is crucial because it transforms the expression into a cleaner and simpler form, making it easier to understand and manipulate. Whenever you encounter a double negative, remember that it's just the same as adding the values!
Understanding this fundamental principle is a helpful step in tackling many mathematical problems.
Irrational Numbers
Irrational numbers like \(\sqrt{3}\) are fascinating! These are numbers that cannot be expressed as a simple fraction of two integers. They go on forever without repeating in their decimal form.
  • Common examples include \(\pi\) and \(e\), along with many square roots of non-perfect squares.
  • The square root of 3, \(\sqrt{3}\), is approximately equal to 1.732, but it never ends and does not repeat, making it irrational.
Irrational numbers are important in mathematics because they fill in the 'gaps' between rational numbers on the number line.
When working with expressions that involve both whole and irrational numbers, like in this exercise, it's crucial to recognize that they cannot be combined into a single, simplified rational expression. This awareness helps keep your calculations accurate.
Whole Numbers
Whole numbers are the set of numbers that start from zero and go on upwards without fractions or decimals. In our exercise, the number 81 is a great example of a whole number.
  • Whole numbers include numbers like 0, 1, 2, 3, and so on.
  • They do not have parts; they are complete, which is why they are called 'whole'.
It's important to recognize whole numbers in expressions because they often provide a stability or reference point from which other types of numbers can be considered, like integers or irrational numbers.
When performing operations like simplification, understanding the role of whole numbers can make the process clearer and allow for more confident problem-solving in mathematics.

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Most popular questions from this chapter

Write as a radical and then simplify. \((-64) 1 / 3\)

Express in radical form. \((1 x)-3 / 5\)

Write as a radical and then simplify. \((-32) 3 / 5\)

Express using rational exponents. $$ 23--\sqrt{4} $$

Perform the operations and simplify. Leave answers in exponential form. \(29 / 221 / 2\)
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