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Chapter 1: Problem 6

Evaluate each expression without using a calculator. $$ \left(\frac{1}{3}\right)^{-2} $$

Short Answer

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Step by step solution

01

Understanding the Concept of Negative Exponents

A negative exponent signifies that instead of multiplying the base, you divide by the base. The expression \( a^{-n} \) can be rewritten as \( \frac{1}{a^n} \). This means that \( \left(\frac{1}{3}\right)^{-2} \) can be converted to its reciprocal with a positive exponent.
02

Convert the Negative Exponent to a Positive Exponent

Rewrite \( \left(\frac{1}{3}\right)^{-2} \) using the rule for negative exponents. This becomes \( \left(\frac{1}{3}\right)^{-2} = \left(\frac{3}{1}\right)^{2} = 3^2 \).
03

Calculate the Power of the Base

Now that we have \( 3^2 \), calculate the power by multiplying 3 by itself: \( 3 \times 3 = 9 \). Therefore, \( 3^2 = 9 \).

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Key Concepts

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

Reciprocal
In mathematics, a reciprocal is simply the inverse of a number. The idea gives the notion that when you multiply a number by its reciprocal, the result is 1. For example, the reciprocal of a fraction, like \( \frac{1}{3} \), is flipping the numerator and the denominator to get \( \frac{3}{1} \) or just 3.
If you're dealing with whole numbers, say 5, the reciprocal is \( \frac{1}{5} \).
  • To find the reciprocal of a fraction: Swap the numerator and the denominator.
  • To find the reciprocal of a whole number: Place it as the denominator over 1.
This concept becomes very useful when dealing with negative exponents, as it helps in turning the fraction the right way around!
Power of a Base
The power of a base is a fundamental idea when it comes to exponents. When we talk about the power of a base, we mean multiplying the base by itself as many times as indicated by the exponent. For instance, in the expression \( 3^2 \), 3 is the base, and 2 is the exponent.
This tells us to multiply 3 by itself: \( 3 \times 3 \). The result of this multiplication gives us what we call "to the power of" something. In this case, that result is 9.
  • Base: The number being multiplied.
  • Exponent: Indicates how many times to multiply the base by itself.
As you can see, the power of a base quickly escalates numbers, especially when exponents are large!
Positive Exponents
Positive exponents are a straight forward concept in math. When you have a positive exponent, it simply means to multiply the base by itself. The power or exponent tells you exactly how many times this multiplication should happen.
For example, \( 3^2 \) means 3 is used as a factor twice, written as \( 3 \times 3 = 9 \).
  • Positive exponents simplify into repeated multiplication.
  • Result is always greater than the base, given the base is greater than 1.
Positive exponents are what normalization often revolves around, especially when converting negative exponents to simpler, more manageable expressions.

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Most popular questions from this chapter

Electronic commerce or e-commerce, buying and selling over the Internet, has been growing rapidly. The total value of U.S. e-commerce in recent years in billions of dollars is given by the exponential function \(f(x)=226(1.11)^{x}\), where \(x\) is the number of years since \(2012 .\) Predict total e-commerce in the year 2020 .

Can the graph of a function have more than one \(x\) -intercept? Can it have more than one \(y\) -intercept?

ENVIRONMENTAL SCIENCE: Wind Energy The use of wind power is growing rapidly after a slow start, especially in Europe, where it is seen as an efficient and renewable source of energy. Global wind power generating capacity for the years 1996 to 2008 is given approximately by \(y=0.9 x^{2}-3.9 x+12.4\) thousand megawatts (MW), where \(x\) is the number of years after \(1995 .\) (One megawatt would supply the electrical needs of approximately 100 homes). a. Graph this curve on the window [0,20] by [0,300] . b. Use this curve to predict the global wind power generating capacity in the year \(2015 .\) [Hint: Which \(x\) -value corresponds to \(2015 ?\) Then use TRACE, VALUE, or TABLE.] c. Predict the global wind power generating capacity in the year \(2020 .\)

How do the graphs of \(f(x)\) and \(f(x+10)+10\) differ?

a. Is the composition of two quadratic functions always a quadratic function? [Hint: Find the composition of \(f(x)=x^{2}\) and \(g(x)=x^{2}\).] b. Is the composition of two polynomials always a polynomial?
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