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Chapter 1: Problem 41

Evaluate each expression without using a calculator. $$ (-8)^{-2 / 3} $$

Short Answer

Expert verified
The value of \((-8)^{-2/3}\) is \(\frac{1}{4}\).

Step by step solution

01

Rewrite the negative exponent

The expression \[ (-8)^{-2/3} \]can be rewritten using the property of negative exponents, which states that any negative exponent can be converted to a positive exponent by taking the reciprocal. So, \[ (-8)^{-2/3} = \frac{1}{(-8)^{2/3}}. \]
02

Understand the meaning of fraction exponents

The fraction exponent \(-2/3\) indicates two operations: the denominator \(3\) suggests we take the cube root, and the numerator \(2\) indicates we then square the result. Therefore, we need to find \[ (-8)^{2/3} = \left((-8)^{1/3}\right)^2. \]
03

Find the cube root

The first step in calculating \((-8)^{2/3}\) is to find the cube root of \(-8\). Since \(-2)^3 = -8\), we find that\[ (-8)^{1/3} = -2. \]
04

Square the result

Next, we square \(-2\) to complete the expression \((-8)^{2/3}\):\[ (-2)^2 = 4. \]
05

Write the reciprocal

Now, substitute back into the reciprocal expression, \((-8)^{-2/3}\) becomes:\[ \frac{1}{(-8)^{2/3}} = \frac{1}{4}. \]

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Key Concepts

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

Negative Exponents
When dealing with negative exponents, it helps to remember a general rule:
  • A negative exponent indicates that we take the reciprocal of the base raised to the corresponding positive exponent.
  • For example, \[ a^{-n} = \frac{1}{a^{n}} \quad \text{for any non-zero } a. \]
In our exercise, we apply this rule to the term \((-8)^{-2/3}\) to transform it into \(\frac{1}{(-8)^{2/3}}.\) This simplifies our calculations by turning the original expression into a fraction with a positive exponent. Remember, handling fractions often makes complex operations more manageable!
Keeping this rule in mind will serve you well across different mathematical contexts, where converting to positive exponents simplifies the process.
Cube Root
The cube root is a special kind of root related to the power of three. It determines a number that, when multiplied by itself twice (three times in total), results in the original number.
  • Mathematically, for a number \( x, \)\( x^{1/3} \)represents the cube root.
  • The cube root of \(-8 \)is found by determining which number cubed gives us \(-8. \)
In this exercise, the cube root of \(-8\)is \(-2,\)since \((-2) \cdot (-2) \cdot (-2) = -8.\)
The practice of finding cube roots is an essential skill in algebra, especially when dealing with fraction powers and simplifying expressions involving roots.
Exponentiation Rules
Exponentiation rules are the backbone of simplifying and solving expressions involving powers. Here are some vital rules:
  • Product of Powers: When multiplying like bases, you add exponents: \( a^{m} \cdot a^{n} = a^{m+n}. \)
  • Power of a Power: When raising a power to another power, you multiply exponents: \( (a^{m})^{n} = a^{m \cdot n}. \)
  • Power of a Product: The exponent applies to each factor within the parentheses: \( (ab)^{n} = a^{n} \cdot b^{n}. \)
In our original problem, this concept guided steps, especially where we worked with fractional exponents.We leveraged these rules to break down \((-8)^{2/3}\)into more manageable parts: first finding the cube root and then squaring the result.
Understanding these rules allows you to navigate through complex exponentiation scenarios with ease and confidence.

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Most popular questions from this chapter

ATHLETICS: Muscle Contraction The fundamental equation of muscle contraction is of the form \((w+a)(v+b)=c,\) where \(w\) is the weight placed on the muscle, \(v\) is the velocity of contraction of the muscle, and \(a, b,\) and \(c\) are constants that depend upon the muscle and the units of measurement. Solve this equation for \(v\) as a function of \(w, a, b,\) and \(c\).

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \sqrt{\frac{9}{x^{4}}} $$

Simplify. $$ \frac{\left(5 x^{2} y^{3} z\right)^{2}}{5(x y z)^{2}} $$

BEHAVIORAL SCIENCES: Smoking and Education According to a study, the probability that a smoker will quit smoking increases with the smoker's educational level. The probability (expressed as a percent) that a smoker with \(x\) years of education will quit is approximately \(y=0.831 x^{2}-18.1 x+137.3\) (for \(10 \leq x \leq 16\) ). a. Graph this curve on the window [10,16] by [0,100] . b. Find the probability that a high school graduate smoker \((x=12)\) will quit. c. Find the probability that a college graduate smoker \((x=16)\) will quit.

GENERAL: Earthquakes The sizes of major earthquakes are measured on the Moment Magnitude Scale, or MMS, although the media often still refer to the outdated Richter scale. The MMS measures the total energy relensed by an earthquake, in units denoted \(M_{W}\) (W for the work accomplished). An increase of \(1 M_{W}\) means the energy increased by a factor of \(32,\) so an increase from \(A\) to \(B\) means the energy increased by a factor of \(32^{B-A}\). Use this formula to find the increase in energy between the following earthquakes: The 2001 earthquake in India that measured \(7.7 M_{W}\) and the 2011 earthquake in Japan that measured \(9.0 M_{W}\). (The earthquake in Japan generated a 28 -foot tsunami wave that traveled six miles inland, killing 24,000 and causing an estimated \(\$ 300\) billion in damage, making it the most expensive natural disaster ever recorded.)
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