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Chapter 7: Problem 34

Evaluate each exponential. $$ 27^{-4 / 3} $$

Short Answer

Expert verified
\(27^{-4/3} = \frac{1}{81}\)

Step by step solution

01

- Rewrite the exponential expression

First, rewrite the exponential expression in a simpler form. Notice that 27 can be written as a power of 3: \[27 = 3^3\]So the expression becomes: \[27^{-4/3} = (3^3)^{-4/3}\]
02

- Apply the power of a power rule

Use the power of a power rule in exponents, which states that \[(a^m)^n = a^{m \times n}\]Applying this rule, we get: \[(3^3)^{-4/3} = 3^{3 \times (-4/3)}\]
03

- Simplify the exponent

Now, simplify the exponent by performing the multiplication: \[3 \times (-4/3) = -4\]This gives us: \[3^{-4}\]
04

- Evaluate the exponential expression

To evaluate \(3^{-4}\), remember that a negative exponent means taking the reciprocal: \[3^{-4} = \frac{1}{3^4}\]Calculate \(3^4\): \[3^4 = 81\]So, the expression becomes: \[3^{-4} = \frac{1}{81}\]

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Key Concepts

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

Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, in the expression \(3^4\), the number 3 is called the base, and the number 4 is the exponent. This means 3 is multiplied by itself four times: \(3 \times 3 \times 3 \times 3 = 81\). Exponents follow specific rules that help us simplify expressions involving them. These rules are crucial to understanding more complex problems.
Power of a Power Rule
The Power of a Power Rule is an essential principle when dealing with exponents. It states that for any base \(a\) and any exponents \(m\) and \(n\), the expression \((a^m)^n\) can be simplified to \(a^{m \times n}\). This rule helps to combine exponents when they are nested. For example, if you have \((2^3)^2\), you multiply the exponents: \(2^{3 \times 2} = 2^6\), making the expression easier to evaluate.
Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the positive exponent. For instance, \(a^{-n}\) equals \(\frac{1}{a^n}\). If you encounter a negative exponent, flip the base to the denominator with a positive exponent. For example, \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\). This rule lets us simplify and make sense of expressions that might initially seem complex.
Simplifying Exponents
Simplifying exponents involves applying various rules to make expressions easier to work with. Here are key steps:
  • Rewrite the bases if possible to make the exponents simpler.
  • Use the power of a power rule to combine exponents.
  • If encountering a negative exponent, convert it using the reciprocal.
For example, simplifying \(27^{-4/3}\) involves rewriting 27 as \(3^3\), applying the power of a power rule to get \((3^3)^{-4/3} = 3^{3 \times (-4/3)} = 3^{-4}\), and then converting the negative exponent to get \(\frac{1}{3^4} = \frac{1}{81}\).

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Most popular questions from this chapter

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{8}}{3-\sqrt{2}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{5 \sqrt{k}}{2 \sqrt{k}+\sqrt{q}} $$

Work each problem. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) because when \(a=1\) and \(b=0,\) a true statement results: $$\sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b$$ Explain why this is faulty reasoning.

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ \frac{-4 \sqrt{13}}{\sqrt{m}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{6}{\sqrt{5}+\sqrt{3}} $$
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