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Chapter 10: Problem 77

Find each power of \(i\) $$ 1^{11} $$

Short Answer

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Step by step solution

01

Understanding Exponents

The number 1 raised to any power remains 1. This is because multiplying 1 by itself any number of times will still yield 1.
02

Solving the Power

Therefore, when calculating \(1^{11}\), we multiply 1 by itself 10 more times:\[1 \times 1 \times 1 \times \ldots \times 1 = 1\]
03

Confirming the Result

Since each multiplication results in 1, the final result of \(1^{11}\) is indeed 1.

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Key Concepts

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

powers of numbers
An exponent, or power, tells you how many times to use a number in a multiplication. For example, in the expression \(1^{11}\), 1 is the base and 11 is the exponent. The base is the number being multiplied, and the exponent tells how many times to multiply the base by itself. When you see an expression like \(1^{11}\), you are looking at **1 raised to the 11th power**. This means you multiply 1 by itself a total of 11 times. However, a unique property of the number 1 is that no matter how many times you multiply it by itself, you still get 1. This concept is not just a math trick but an essential rule that applies to any base of 1 raised to any power.
multiplying by itself
Multiplying a number by itself is a simple concept at the heart of understanding exponents. When you have a number like 1, and you're asked to calculate \(1^{11}\), you're technically performing the operation:
  • 1 x 1 = 1
  • 1 x 1 = 1
  • ... continue this process ...
until you've multiplied 1 a total of 11 times. Even though multiplying 1 over and over seems redundant because it doesn't change the result, this repetitive multiplication helps reinforce the idea of what exponents represent. For bases other than 1, this process might result in a product much larger or smaller than the original base, showing the power of multiplying by itself.
basic exponent rules
Learning basic exponent rules can simplify many mathematical calculations and problems. Here are some key rules:
  • Any number raised to the power of 1 is the number itself: \(a^1 = a\).
  • Any number raised to the power of 0 is 1, provided the base is not 0: \(a^0 = 1\).
  • When multiplying like bases with exponents, you add the exponents: \(a^m \times a^n = a^{m+n}\).
Another fundamental rule is that any number raised to a power results in multiplying the base by itself the specified number of times, such as \(a^n = a \times a \times ... \times a\), \(n\) times. Understanding these rules helps tackle more challenging exponent problems and further builds up your arithmetic skills.

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Most popular questions from this chapter

The number of cellular telephone subscriptions in the United States from 1996 through 2006 can be modeled by the function \(f(x)=33.3 x^{45},\) where \(y\) is the number of cellular telephone subscriptions in millions, \(x\) years after \(1996 .\) (Source: Based on data from the Cellular Telecommunications \& Internet Association, \(1994-2000)\) Use this information to answer Exercises. Use this model to estimate the number of cellular telephone subscriptions in the United States in \(2006 .\) Round to the nearest tenth of a million.

The Mosteller formula for calculating adult body surface area is \(B=\sqrt{\frac{h w}{3131}},\) where \(B\) is an individual's body surface area in square meters, \(h\) is the individual's height in inches, and \(w\) is the individual's weight in pounds. Use this information to answer Exercises 109 and \(110 .\) Round answers to 2 decimal places. Find the body surface area of an individual who is 66 inches tall and who weighs 135 pounds.

Identify the domain and then graph each function. $$f(x)=\sqrt{x}-2$$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$g(-19)$$

Which of the following are not real numbers? $$\sqrt[3]{-17}$$
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