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Chapter 1: Problem 24

Evaluate each expression without using a calculator. $$ (-27)^{2 / 3} $$

Short Answer

Expert verified
The value of \[ (-27)^{2/3} \] is 9.

Step by step solution

01

Understand the Problem

The expression \[ (-27)^{2/3} \]involves a negative number raised to a fractional power. The fractional exponent \( \frac{2}{3} \) indicates taking the cube root followed by squaring.
02

Simplify the Fractional Exponent

Rewrite the expression \[ (-27)^{2/3} \]as \[ \left( (-27)^{1/3} \right)^2 \].This represents the cube root of \(-27\) raised to the 2nd power.
03

Calculate the Cube Root

Find the cube root of \(-27\), which is \[ (-27)^{1/3} = -3 \],because \[ (-3)^3 = -27 \].
04

Square the Result

Take the result from the cube root and square it:\[ (-3)^2 = 9 \].

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Key Concepts

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

Cube Roots
Cube roots are a way to reverse cube operations. Thinking about it, a cube root helps us find a number which, when multiplied by itself three times, gives the original number. For instance, the cube root of \(-27\) is \-3\. This is because \((-3) \times (-3) \times (-3) = -27\). Cube roots are especially interesting because they can involve negative numbers, as cube operations involving negative numbers still produce negative results.

When dealing with cube roots, here are some things to remember:
  • If a number is negative, like \(-27\), its cube root will also be negative.
  • The symbol for cube root is often represented with an exponent of \(1/3\)
  • Cube roots are one of the simplest roots because they work with both positive and negative numbers easily.
Negative Numbers in Exponents
Exponents with negative numbers can sometimes be surprising. When we talk about exponents, we're often thinking about multiplying a number by itself. But what happens when that number is negative? Let's think of \(-27\). If we raise this number to a power, like in the case \((-27)^{2/3}\), the negative base here is well-managed because the exponent \(2/3\) tells us to perform operations, like taking a cube root and then squaring, separately.

Here are some helpful pointers:
  • If the exponent involves a fraction like \(2/3\), solve in parts 鈥� take the root first, then raise to the remaining power.
  • Don't forget that if a number is negative, its cube operations generally still result in negative numbers.
  • Be mindful that other operations can change negativity depending on how the power parts are handled, especially squaring.
Exponentiation Rules
Exponentiation rules help make complex problems easier by applying specific operations. When dealing with fractional exponents like \(\frac{2}{3}\), it's really about understanding the separate parts involved: the root and the power. Thinking of the expression \((-27)^{2/3}\) invites these key exponent rules:

  • First, break down the exponent: \(a^{m/n}\) means take the n-th root of \(a\) and then raise the result to the m-th power.
  • In this case, calculate the cube root first, turning \((-27)^{1/3}\) to \(-3\). Afterwards, you square the result: \((-3)^2\) equals 9.
  • Remember, squaring a number makes it positive, hence the final result becomes positive even if the base was initially negative.
By fluently applying these rules, we grow to manage challenging problems with ease, focusing on operations one step at a time.

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Most popular questions from this chapter

93 -94. ALLOMETRY: Heart Rate It is well known that the hearts of smaller animals beat faster than the hearts of larger animals. The actual relationship is approximately (Heart rate) \(=250(\text { Weight })^{-1 / 4}\) where the heart rate is in beats per minute and the weight is in pounds. Use this relationship to estimate the heart rate of: A 16 -pound \(\operatorname{dog}\).

BUSINESS: The Rule of .6 Many chemical and refining companies use "the rule of point six" to estimate the cost of new equipment. According to this rule, if a piece of equipment (such as a storage tank) originally cost C dollars, then the cost of similar equipment that is \(x\) times as large will be approximately \(x^{06} \mathrm{C}\) dollars. For example, if the original equipment cost \(C\) dollars, then new equipment with twice the capacity of the old equipment \((x=2)\) would cost \(2^{0.6} \mathrm{C}=1.516 \mathrm{C}\) dollars - that is, about 1.5 times as much. Therefore, to increase capacity by \(100 \%\) costs only about \(50 \%\) more. \({ }^{*}\) Use the rule of .6 to find how costs change if a company wants to triple \((x=3)\) its capacity.

a. Find the composition \(f(g(x))\) of the two linear functions \(f(x)=a x+b\) and \(g(x)=c x+d \quad\) (for constants \(a, b, c,\) and \(d)\) b. Is the composition of two linear functions always a linear function?

ATHLETICS: Juggling If you toss a ball \(h\) feet straight up, it will return to your hand after \(T(h)=0.5 \sqrt{h}\) seconds. This leads to the juggler's dilemma: Juggling more balls means tossing them higher. However, the square root in the above formula means that tossing them twice as high does not gain twice as much time, but only \(\sqrt{2} \approx 1.4\) times as much time. Because of this, there is a limit to the number of balls that a person can juggle, which seems to be about ten. Use this formula to find: a. How long will a ball spend in the air if it is tossed to a height of 4 feet? 8 feet? b. How high must it be tossed to spend 2 seconds in the air? 3 seconds in the air?

ENVIRONMENTAL SCIENCE: Wind Energy The use of wind power is growing rapidly after a slow start, especially in Europe, where it is seen as an efficient and renewable source of energy. Global wind power generating capacity for the years 1996 to 2008 is given approximately by \(y=0.9 x^{2}-3.9 x+12.4\) thousand megawatts (MW), where \(x\) is the number of years after \(1995 .\) (One megawatt would supply the electrical needs of approximately 100 homes). a. Graph this curve on the window [0,20] by [0,300] . b. Use this curve to predict the global wind power generating capacity in the year \(2015 .\) [Hint: Which \(x\) -value corresponds to \(2015 ?\) Then use TRACE, VALUE, or TABLE.] c. Predict the global wind power generating capacity in the year \(2020 .\)
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