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Chapter 3: Problem 47

Expand and simplify each of the following. $$\left(-\frac{2}{3}\right)^{3}$$

Short Answer

Expert verified
The expanded and simplified form is \( \frac{-8}{27} \).

Step by step solution

01

Understand the Problem

The exercise asks us to expand and simplify the expression \(\left(-\frac{2}{3}\right)^{3}\). This means we'll raise \(-\frac{2}{3}\) to the power of 3.
02

Apply the Power Rule

When a fraction is raised to a power, we raise both the numerator and the denominator to the power separately. In this case, we need to calculate \(\left(-\frac{2}{3}\right)^{3} = \frac{(-2)^{3}}{3^{3}}\).
03

Calculate the Cubes

Calculate \((-2)^{3}\) and \(3^{3}\):- \((-2)^{3} = (-2)\times(-2)\times(-2) = -8\)- \(3^{3} = 3\times3\times3 = 27\)Thus, \(\frac{(-2)^{3}}{3^{3}} = \frac{-8}{27}\).
04

Simplify the Expression

The fraction \(\frac{-8}{27}\) is already in its simplest form, as the numerator and the denominator have no common factors other than 1. Thus, the expanded and simplified form of \(\left(-\frac{2}{3}\right)^{3}\) is \(\frac{-8}{27}\).

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Key Concepts

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

Fraction Exponents
Fraction exponents, often referred to as rational exponents, are a way of representing powers and roots together. For example, when we see a fraction exponent like \(a^{m/n}\), it indicates both an exponentiation and a root operation.

  • The numerator \(m\) in the fraction \(m/n\) indicates the power to which the base \(a\) is raised.
  • The denominator \(n\) indicates the root that is taken of the result.

Fractional exponents are useful in simplifying expressions that involve both roots and powers. Instead of using the traditional radical signs, they provide a more straightforward notation. For example, \(a^{1/2}\) is equivalent to the square root of \(a\), or \(\sqrt{a}\). By using fractional exponents, calculations can often be performed more efficiently.
Power Rule
The power rule in exponentiation is crucial when dealing with expressions raised to a power. This rule states that when a fraction (or any number) is raised to a whole number power, you apply the power to both the numerator and the denominator separately.

For an expression \(\left(\frac{a}{b}\right)^n\), the power rule dictates:
  • The numerator is calculated as \(a^n\).
  • The denominator is calculated as \(b^n\).

In the example \(\left(-\frac{2}{3}\right)^3\), we used the power rule by computing \((-2)^3\) and \(3^3\). This method ensures the expression is handled systematically and helps avoid common calculation mistakes.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This step is often the final part of solving an exponentiation problem. The goal is to ensure the expression is easy to interpret, often involving reducing fractions and eliminating any unnecessary complexity.

  • Ensure that the numbers in the fraction do not have any common factors other than 1.
  • This means confirming that the greatest common divisor (GCD) of the numerator and the denominator is 1.

In our example, the fraction \(-\frac{8}{27}\) was checked for simplification. Both 8 and 27 do not share any common factors other than 1. Therefore, it is already in its simplest form. Simplifying is crucial as it often provides the most elegant and interpretable version of the expression.

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Most popular questions from this chapter

Reduce each fraction to lowest terms. $$\frac{210}{462}$$

Suppose 33 people took the final exam in a math class. If 11 people got an \(\mathrm{A}\) on the final exam, what fraction of the students did not get an A on the exam? Write your answer in lowest terms.

Reduce each fraction to lowest terms. $$\frac{180 x y z}{108 x y}$$

Add and subtract the following mixed numbers as indicated. $$\begin{array}{r}9 \frac{11}{12} \\\\+4 \frac{1}{6} \\\\\hline\end{array}$$

Simplify. $$9 \cdot 6+5$$
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