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Chapter 5: Problem 24

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

Short Answer

Expert verified
\(\frac{1}{27}\)

Step by step solution

01

Evaluate the Inner Exponent

Start by evaluating the innermost exponent. Here, the base is 3 and the exponent is -1, which can be written as the reciprocal: \[ 3^{-1} = \frac{1}{3} \]
02

Apply the Outer Exponent

Now, apply the outer exponent to the result from Step 1. Since we have \(\left(\frac{1}{3}\right)^{3}\), this means multiplying the fraction by itself three times:\[ \left(\frac{1}{3}\right)^{3} = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \]
03

Calculate the Result

Multiply the fractions: First, calculate the numerator: \(1 \times 1 \times 1 = 1\).Next, calculate the denominator: \(3 \times 3 \times 3 = 27\).So, the expression simplifies to:\[ \frac{1}{27} \]

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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 seem a bit tricky at first, but with a simple rule, they become easy to handle. When you see a negative exponent, it means you need to take the reciprocal of the base and make the exponent positive. For example, if you have an expression like \( 3^{-1} \), this doesn't mean you'd have a negative number but instead the reciprocal, \( \frac{1}{3} \). This rule holds true for any non-zero base \( a \), where \( a^{-n} = \frac{1}{a^n} \). By understanding this, you'll find that evaluating expressions with negative exponents becomes straightforward.
It's essential to get comfortable with this concept, as it is quite common in algebra and calculus.
Fraction Multiplication
Multiplying fractions is simpler than you might think. When you multiply fractions, you don't cross-multiply. Instead, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together directly. For example, if you have the expression \( \left(\frac{1}{3}\right)^3 \), this means you multiply \( \frac{1}{3} \) by itself three times: \( \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \).
  • Multiply the numerators: \( 1 \times 1 \times 1 = 1 \)
  • Multiply the denominators: \( 3 \times 3 \times 3 = 27 \)
The result is \( \frac{1}{27} \). So, multiplying fractions follows a straightforward path where you handle numerators and denominators separately. Remembering this process will make fraction multiplication easy to tackle.
Numerical Expressions
A numerical expression involves numbers and operations without an equal sign. Simplifying numerical expressions means you perform these operations to reach a single number. For instance, in the expression \( \left(3^{-1}\right)^3 \), we start by simplifying the exponential parts first. This entails dealing with negative exponents and applying them step by step, as shown in the detailed solution.
Performing operations systematically ensures accuracy. When simplifying, always:
  • Resolve exponents first, including handling negative exponents by taking reciprocals
  • Follow by completing any multiplication or division as seen in the expression
By breaking down the process, complex-looking numerical expressions become much easier to handle in bite-sized steps.

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Most popular questions from this chapter

Use scientific notation and the properties of exponents to help you perform the following operations. \(\frac{0.000064}{16,000}\)

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(-\sqrt[5]{(x-y)^{2}}\)

Use scientific notation and the properties of exponents to help you perform the following operations. \((0.00007)(11,000)\)

In the year 2000 the public debt of the United States was approximately \(\$ 5,700,000,000,000\). For July 2000 , the census reported that \(275,000,000\) 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.

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{18 x^{\frac{1}{3}}}{9 x^{\frac{1}{4}}}\right)^{2}\)
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