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Chapter 5: Problem 21

Simplify each numerical expression. \(\left(3^{-1}\right)^{-3}\)

Short Answer

Expert verified
The simplified expression is 27.

Step by step solution

01

Apply Power of a Power Rule

The expression given is \( \left(3^{-1}\right)^{-3} \). The power of a power rule states that \( (a^m)^n = a^{m \cdot n} \). Apply this rule here to get \( 3^{-1 \cdot (-3)} = 3^{3} \).
02

Calculate the Result

Now we have \( 3^{3} \). Calculate this by multiplying three times: \( 3 \times 3 = 9 \), and then \( 9 \times 3 = 27 \).

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Key Concepts

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

Power of a Power Rule
The power of a power rule is a technique used in exponentiation. This rule helps simplify expressions where there is an exponent raised to another exponent. It's a useful rule that makes complex expressions more manageable.To understand this rule more deeply, let's break it down into a simple format:
  • If you have an expression like \((a^m)^n\), it means you multiply the exponents together, resulting in \(a^{m \cdot n}\).
  • This rule allows you to change the order of operations in exponentiation without changing the result.
  • In our example with \( (3^{-1})^{-3} \), applying the power of a power rule gives us \(3^{-1 \cdot (-3)} = 3^3\).
The minus signs in exponents significantly affect the outcome, transforming a complex inner need for negative powers into a simple positive exponent. This allows us to work with the expression as a regular power.
Simplification
Simplification is the process of reducing expressions to their most basic form. When simplifying, we aim to make the expression as straightforward as possible.For example, in our given expression, we started with \( (3^{-1})^{-3} \). Applying the power of a power rule reduced it to a simpler expression: \(3^3\).
  • The primary goal is to perform operations in a step-by-step manner.
  • Continuing simplification, \(3^3\) simplifies to 27 by performing the multiplication: \(3 \times 3 = 9\) and then \(9 \times 3 = 27\).
By following these steps, what might seem complex at first becomes simple and easy to handle. Always aim for clarity and simplicity in simplification to make further calculations easier.
Numerical Expressions
Understanding numerical expressions is crucial in mathematics, as it involves calculations with numbers exclusively. By recognizing patterns and using rules like the power of a power rule, simplifying these expressions becomes more intuitive.Numerical expressions, such as \( (3^{-1})^{-3} \), often involve:
  • Numbers that need to be calculated or reduced to a simplified form.
  • Understanding and applying rules properly helps solve them efficiently.
  • Expressing complicated ideas in a series of simple steps allows you to reach a solution quickly.
Working through numerical expressions is about finding patterns and applying logical thinking to reach the correct numerical answer. Developing these skills enhances problem-solving capabilities and builds a strong foundation in mathematics.

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Most popular questions from this chapter

Your friens keeps getting an error message when evaluating \(-4^{\frac{5}{2}}\) on his calculator. What error is he probably making?

Simplify each of the following. Express final results using positive exponents only. $$ \left(3 x^{\frac{1}{4}} y^{\frac{1}{5}}\right)^{3} $$

For Problems \(1-18\), write each of the following in scientific notation. \(0.00000082\)

In the year 2008 the public debt of the United States was approximately $$\$ 10,600,000,000,000$$. For July 2008 , the census reported that \(303,000,000\) people lived in the United States. Convert these figures to scientific notation, and compute the average debt per person. Express the result in scientific notation.

Use your calculator to evaluate each of the following. Express final answers in ordinary notation. (a) \(27,000^{2}\) (b) \(450,000^{2}\) (c) \(14,800^{2}\) (d) \(1700^{3}\) (e) \(900^{4}\) (f) \(60^{5}\) (g) \(0.0213^{2}\) (h) \(0.000213^{2}\) (i) \(0.000198^{2}\) (j) \(0.000009^{3}\)
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