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Chapter 5: Problem 54

Evaluate each expression. $$ \frac{1}{3^{-3}} $$

Short Answer

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Step by step solution

01

Understand the expression

The given expression is \( \frac{1}{3^{-3}} \). It involves a fraction with a base and an exponent in the denominator.
02

Simplify the exponent

To simplify \( 3^{-3} \), recall that a negative exponent means taking the reciprocal of the base raised to the positive exponent: \[ 3^{-3} = \frac{1}{3^3} \].
03

Simplify the fraction

Substitute \( 3^{-3} \) with \( \frac{1}{3^3} \) in the original expression: \[ \frac{1}{3^{-3}} = \frac{1}{\frac{1}{3^3}} \].
04

Invert the fraction

To divide by a fraction, multiply by its reciprocal: \[ \frac{1}{\frac{1}{3^3}} = 3^3 \].
05

Calculate the power

Calculate \( 3^3 \): \[ 3^3 = 3 \times 3 \times 3 = 27 \].

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Key Concepts

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

Negative Exponents
Negative exponents might look a bit scary at first, but they are actually very manageable! Let's break it down. A negative exponent indicates that you need to take the reciprocal of the base and then raise it to the positive version of that exponent.
For example, if you have \(a^{-n}\), it can be rewritten as \( \frac{1}{a^n}\). This means you flip the base into a fraction and change the exponent to a positive.
For instance, consider the expression \(3^{-3}\). To simplify this:
  • First, acknowledge that the exponent is negative, which means you take the reciprocal of 3.
  • Next, raise it to the power of 3: \(3^{-3} = \frac{1}{3^3}\).
Negative exponents can turn seemingly complex expressions into simple fractions, making them easier to handle.
Reciprocals
Understanding reciprocals is crucial in simplifying expressions with exponents, especially negative ones. The reciprocal of a number is essentially 'flipping' it. If the number is a fraction, you swap its numerator and denominator.
For example, the reciprocal of 5 is \(\frac{1}{5}\), and the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). In simpler terms, if \(a\) is a number, its reciprocal is \( \frac{1}{a} \).
When dealing with negative exponents, think in terms of reciprocals:
  • For instance, \(3^{-3}\) is the reciprocal of \(3^3\).
  • This changes the original fraction \( \frac{1}{3^{-3}} \) to \( \frac{1}{\frac{1}{3^3}} \).
To divide by a fraction, you multiply by its reciprocal. That's why the expression \( \frac{1}{\frac{1}{3^3}} \) simplifies to \( 3^3 \).
Powers of Numbers
Powers, also known as exponents, tell you how many times to multiply a number by itself. It's like repeated multiplication.
For example, \(3^3\) means you multiply 3 by itself three times: \(3 \times 3 \times 3\). The result is 27.
When calculating powers, especially with whole numbers, follow these steps:
  • Write down the base number.
  • Multiply this number by itself as many times as the exponent indicates.
  • So, for \(3^3\), multiply 3 by itself twice more: \(3 \times 3 = 9\) and then \(9 \times 3 = 27\).
Once you understand this, dealing with larger exponents or different bases becomes easier!

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