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Chapter 0: Problem 28

Compute the numbers. $$1^{-1.2}$$

Short Answer

Expert verified
The result of \(1^{-1.2}\) is 1.

Step by step solution

01

Understanding the Expression

The expression given is \(1^{-1.2}\). This means we need to find the result of 1 raised to the power of -1.2.
02

Apply Negative Exponent Rule

Recall the rule for negative exponents: \(a^{-b} = \frac{1}{a^b}\). Here, \(a = 1\) and \(b = 1.2\).
03

Simplify the Expression

Since any non-zero number raised to any power is still 1, \(1^{1.2}\) equals 1. Therefore, \(\frac{1}{1^{1.2}} = \frac{1}{1} = 1\).

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Key Concepts

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

Negative Exponents
Negative exponents can seem confusing at first, but they're quite straightforward once you understand the rules. When you see a negative exponent, like in the expression \(a^{-b}\), it means that you take the reciprocal of the base raised to the positive value of the exponent.
For example:
\(2^{-3} = \frac{1}{2^{3}} = \frac{1}{8}\).
This rule helps simplify calculations and is essential in various areas of math and science.
In our original problem, we used the negative exponent rule on 1, giving us: \(1^{-1.2} = \frac{1}{1^{1.2}} = 1\).
Since any non-zero number raised to any power is still itself, it becomes very simple to compute.
Exponentiation
Exponentiation is a fundamental operation in mathematics that deals with numbers raised to a power. It is written as \(a^b\), where \(a\) is the base and \(b\) is the exponent. The exponent indicates how many times you multiply the base by itself.
For instance:
\(2^3 = 2 \cdot 2 \cdot 2 = 8\).
In the given problem, the base is 1 and the exponent is -1.2. Before applying the negative exponent rule, we should understand that:
1 raised to any power is still 1, such as
\(1.2 = 1 \cdot 1 \cdot 1.2 \cdot 1 = 1\).
Exponentiation is a critical concept for progressing into more advanced mathematical functions and understanding how numbers interact under various operations.
Mathematical Rules
Mathematical rules provide the necessary structure and clarity for solving expressions and equations. These rules like the order of operations, exponent rules, and properties of numbers help us consistently get correct results.
Key rules include:
  • Order of Operations: Follow PEMDAS/BODMAS (Parentheses/Brackets, Orders (exponents), Multiplication and Division, Addition and Subtraction).
  • Exponent Rules: Fundamental rules like \(a^m \cdot a^n = a^{(m+n)}\) and \( (a^m)^n = a^{(mn)}\).
  • Negative Exponents: As highlighted earlier, \(a^{-m} = \frac{1}{a^m}\).

In the original exercise, understanding these principles, especially the negative exponent rule, is what allows us to solve the question effectively. So, knowing these mathematical rules helps us tackle various problems accurately and efficiently.

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