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Chapter 8: Problem 80

Find each power of i. $$ 1^{102} $$

Short Answer

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Step by step solution

01

Understanding the problem

The given problem requires finding the power of 1 raised to 102. The formula to find powers of 1 is straightforward as any number raised to any power will remain the same number.
02

Simplifying the power

Since the base is 1, any exponent on 1 will yield 1. This means: \[ 1^{102} = 1 \]

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Key Concepts

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

Exponents
Exponents are a way to represent repeated multiplication of a number by itself. When you see something like \(a^n\), it means you multiply 'a' by itself 'n' times. For example, \(2^3 = 2 \times 2 \times 2 = 8\). Exponents can make numbers grow very quickly, but they also follow certain rules that make them easier to work with.
  • The base in \(a^n\) is 'a'鈥攖he number being multiplied.
  • The exponent is 'n'鈥攖elling us how many times to multiply the base by itself.

In our original exercise, the base was 1, and the exponent was 102. But there's a special property when dealing with 1 as the base, which leads us to our next section.
Properties of Exponents
Understanding how exponents behave can simplify many calculations. There are several basic properties:
  • Any number raised to the power of 1 remains the number: \(a^1 = a\)
  • Any number raised to the power of 0 is 1: \(a^0 = 1\) (as long as \(a e 0)\)
  • Multiplying like bases: \(a^m \times a^n = a^{m+n} \)
  • Dividing like bases: \(a^m / a^n = a^{m-n} \) when \(m > n\)
  • Raising a power to a power: \((a^m)^n = a^{m\cdot n} \)
  • Multiplying different bases but same exponent: \(a^n \times b^n = (a \times b)^n \)

In our example, we were dealing with 1 as the base. Any non-zero number raised to any power remains itself. Therefore, \(1^{102} = 1\). This leads us to the final part.
Simplifying Expressions
Simplifying expressions with exponents can make otherwise complicated problems much easier to solve. Here are a few tips:
  • Look for patterns: Often, you'll find that certain expressions repeat or follow a clear rule.
  • Break down the problem: If the expression seems complex, try breaking it down using the properties of exponents we discussed earlier.
  • Combine like terms: Expressions with the same base can often be combined or simplified.

In the given problem, simplifying was straightforward because a base of 1 raised to any power is always 1. Knowing the properties of exponents, we quickly arrive at \(1^{102} = 1\). By understanding and applying these core concepts, you can solve exponent problems more efficiently and accurately.

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Most popular questions from this chapter

Simplify by first writing the radicals as radicals with the same index. Then multiply. Assume that all variables represent positive real numbers. $$ \sqrt[5]{7} \cdot \sqrt[7]{5} $$

Rationalize each denominator. Assume that all radicals represent real numbers and no denominators are 0. $$ \frac{q}{\sqrt{5+q}} $$

Find the distance between each pair of points. $$ (-6,5) \text { and }(3,-4) $$

What is the conjugate of \(a+b i ?\)

The following expression occurs in a certain standard problem in trigonometry. $$ \frac{\sqrt{3}+1}{1-\sqrt{3}} $$ Show that it simplifies to \(-2-\sqrt{3} .\) Then verify, using a calculator approximation.
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