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Chapter 2: Problem 5

Evaluate the given expression. Do not use a calculator. $$ \left(\frac{2}{3}\right)^{-4} $$

Short Answer

Expert verified
The given expression can be rewritten as \(\frac{1}{\left(\frac{2}{3}\right)^{4}}\). Simplifying further, it becomes \(\frac{1}{\frac{16}{81}}\), which is equal to \(\frac{81}{16}\).

Step by step solution

01

Apply the negative exponent rule

We will use the negative exponent rule to convert the expression into a form that is easier to work with. The rule states that \(a^{-n} = \frac{1}{a^n}\). Applying this to our expression, we get: \[ \left(\frac{2}{3}\right)^{-4} = \frac{1}{\left(\frac{2}{3}\right)^{4}} \]
02

Simplify the expression

Now, we need to simplify the expression by raising the fraction to the power of 4: \[ \frac{1}{\left(\frac{2}{3}\right)^{4}} = \frac{1}{\left(\frac{2^4}{3^4}\right)} \]
03

Calculate the powers

Next, we will calculate \(2^4\) and \(3^4\) to obtain: \[ \frac{1}{\left(\frac{2^4}{3^4}\right)} = \frac{1}{\left(\frac{16}{81}\right)} \]
04

Invert and multiply the fractions

To complete the evaluation, we will invert the fraction in the denominator and multiply: \[ \frac{1}{\frac{16}{81}} = \frac{1}{1} \times \frac{81}{16} = \frac{81}{16} \] The expression \(\left(\frac{2}{3}\right)^{-4}\) evaluates to \(\frac{81}{16}\).

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Key Concepts

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

Understanding Fraction Exponents
Fraction exponents, also known as rational exponents, can be a bit confusing at first. They involve both the concepts of roots and powers.
When you see an exponent that is a fraction, such as \(a^{m/n}\), it can actually be translated into radicals and powers as \((a^m)^{1/n}\) or \(\sqrt[n]{a^m}\).
For instance, in our expression \(\left(\frac{2}{3}\right)^{-4}\), the exponent \(-4\) tells us not only to take the power of 4 but also to consider the inverse, as guided by the negative sign. The key feature of fraction exponents is to bridge the operation of raising a number to a power and taking its root, a process that can help simplify many algebraic problems.
Simplifying Expressions with Negative Exponents
Simplifying expressions is a crucial skill, especially when dealing with exponents. The negative exponent rule states: \(a^{-n} = \frac{1}{a^n}\).
This means we invert the base number and then apply the positive power. Applying this to \(\left(\frac{2}{3}\right)^{-4}\), first, we use the negative exponent rule, turning it into \(\frac{1}{\left(\frac{2}{3}\right)^4}\).
Next, we simplify further by raising the fraction to this power. This involves calculating \(\left(\frac{2}{3}\right)^{4} = \frac{2^4}{3^4}\), leading to \(\frac{1}{\frac{16}{81}}\).
Each step of simplification focuses on reducing the complexity of the expression while adhering to foundational rules of algebra.
The Magic of Inverting Fractions
Inverting fractions is a nifty trick that comes into play when fractions are in the denominator.
When you have a fraction like \(\frac{1}{\frac{a}{b}}\), it is equivalent to multiplying by the reciprocal of that fraction, turning \(\frac{1}{\frac{a}{b}} = b/a\).
In our original problem, once we calculate \(\frac{1}{\frac{16}{81}}\), we invert \(\frac{16}{81}\) to \(\frac{81}{16}\).
This flips the fraction upside down, turning it into a straightforward multiplication operation, thus further simplifying the expression. Understanding this technique helps immensely in dealing with more complex fractions and is often a key simplification step in algebraic fraction operations.

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Most popular questions from this chapter

Show that \((99+70 \sqrt{2})^{1 / 3}=3+2 \sqrt{2}\).

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\) $$ f(x)=-\frac{3}{x} $$

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\) $$ f(x)=-\frac{3}{x}+4 $$

Using the result that \(\sqrt{2}\) is irrational (proved in Section 0.1\()\), show that \(2^{5 / 2}\) is irrational.

Show that \(3^{3 / 2} 12^{3 / 2}=216\)
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