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Chapter 4: Problem 15

Evaluate: a. \(27^{2 / 3}\) b. \(16^{-3 / 4}\) c. \(25^{-3 / 2}\) d. \(81^{-34}\)

Short Answer

Expert verified
a. 9, b. \frac{1}{8}, c. \frac{1}{125}, d. \frac{1}{81^{34}} or practical 0.

Step by step solution

01

Simplify the first expression

Evaluate the expression **a.** \(27^{2/3}\)The base of 27 can be written as \(3^3\), thus we can simplify:\(27^{2/3} = (3^3)^{2/3}\)Using the power of a power property \((a^m)^n = a^{m \times n}\):\((3^3)^{2/3} = 3^{3 \times \frac{2}{3}} = 3^2 = 9\)
02

Simplify the second expression

Evaluate the expression **b.** \(16^{-3/4}\)16 can be written as \(2^4\), thus we can simplify:\(16^{-3/4} = (2^4)^{-3/4}\)Using the power of a power property and negative exponent property \((a^m)^{-n} = a^{-mn}\):\((2^4)^{-3/4} = 2^{4 \times -\frac{3}{4}} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
03

Simplify the third expression

Evaluate the expression **c.** \(25^{-3/2}\)The base of 25 can be written as \(5^2\), thus we can simplify:\(25^{-3/2} = (5^2)^{-3/2}\)Using the power of a power property and negative exponent property:\((5^2)^{-3/2} = 5^{2 \times -\frac{3}{2}} = 5^{-3} = \frac{1}{5^3} = \frac{1}{125}\)
04

Simplify the fourth expression

Evaluate the expression **d.** \(81^{-34}\)Using the property of integer exponents, any number to the power of zero is 1:\(81^{-34} = (81^{34})^{-1} = \frac{1}{81^{34}}\).Since the exact value is too large to express here, we summarize as \(81^{-34}\) approaches zero (or practically equivalent to zero).

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Key Concepts

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

Power of a Power Property
The **power of a power** property is a fundamental rule in exponentiation. It states that when an exponent is raised to another exponent, you multiply the exponents together. For example, \( (a^m)^n = a^{m \times n} \). This property simplifies complex expressions and makes calculations easier. Let's apply it:

For the expression \( 27^{2/3} \):
  • First, express 27 as \( 3^3 \).
  • Then, using the power of a power property, \( (3^3)^{2/3} = 3^{3 \times 2/3} = 3^2 \).
  • This simplifies to \( 9 \).
Understanding this property helps in breaking down and simplifying many exponential expressions you will encounter in math.
Negative Exponents
Negative exponents might seem confusing at first, but they simplify expressions greatly. A **negative exponent** indicates that the base should be taken as a reciprocal. The rule can be written as \[ a^{-n} = \frac{1}{a^n} \]. Let's look at an example:

For the expression \( 16^{-3/4} \):
  • First, express 16 as \( 2^4 \).
  • Using the power of a power property, write \( (2^4)^{-3/4} = 2^{4 \times -3/4} = 2^{-3} \).
  • Then apply the negative exponent rule: \( 2^{-3} = \frac{1}{2^3} \).
  • This simplifies to \( \frac{1}{8} \).
Recognizing and applying these rules makes working with negative exponents straightforward and helps in simplifying complex expressions.
Simplifying Expressions
Simplifying expressions involves breaking them down to their simplest form using exponent rules. Knowing the properties of exponents, including **power of a power** and **negative exponents**, is essential. Let鈥檚 go through an example:

For the expression \( 25^{-3/2} \):
  • Express 25 as \( 5^2 \).
  • Using the power of a power property, write \( (5^2)^{-3/2} = 5^{2 \times -3/2} = 5^{-3} \).
  • Apply the negative exponent rule: \( 5^{-3} = \frac{1}{5^3} \).
  • This simplifies to \( \frac{1}{125} \).
Similarly, for larger exponents, the patterns follow:
For the expression \( 81^{-34} \):
  • By applying the negative exponent rule: \( 81^{-34} = \frac{1}{81^{34}} \).
  • Since \( 81^{34} \) is an incredibly large number, the value of \( \frac{1}{81^{34}} \) is practically close to zero.
Mastering these steps and rules will help you simplify even the most complex expressions.

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Most popular questions from this chapter

Determine if the expressions are true or false. If false, change the right- hand side to make the expression true. a. \(0.00756=7.56 \cdot 10^{-2}\) b. \(3.432 \cdot 10^{5}=343,200\) c. 49 megawatts \(=4.9 \cdot 10^{6}\) watts d. \(1,596,000,000=1.5 \cdot 10^{9}\) e. 5 megapixels \(=5.0 \cdot 10^{6}\) pixels f. 6 picoseconds \(=6.0 \cdot 10^{12}\) seconds

Use \(<,>,\) or \(=\) to make each statement true. a. \(\sqrt{3}+\sqrt{7} \underline{?} \sqrt{3+7}\) b. \(\sqrt{3^{2}+2^{2}} \underline{?} 5\) c. \(\sqrt{5^{2}-4^{2}} \geq 2\)

Change each number to scientific notation, then simplify using rules of exponents. Show your work, recording your final answer in scientific notation. a. \(10 \%\) of 0.00001 b. \(\frac{0.00005}{50,000}\) c. \(\frac{3}{0.006}\) d. \(\frac{8000}{0.0008}\) e. \(\frac{0.0064}{8000}\) f. \(5,000,000 \cdot 40,000\)

(Requires a calculator that can evaluate powers.) Calculators and spreadsheets use slightly different formats for scientific notation. For example, if you type in Avogadro's number either as 602,000,000,000,000,000,000,000 or as \(6.02 \cdot 10^{23},\) the calculator or spreadsheet will display \(6.02 \mathrm{E} 23,\) where \(\mathrm{E}\) stands for "exponent" or power of \(10 .\) Perform the following calculations using technology, then write the answer in standard scientific notation rounded to three places. a. \(\left(\frac{9}{5}\right)^{50}\) b. \(2^{35}\) c. \(\left(\frac{1}{3}\right)^{7}\) d. \(\frac{7}{6^{15}}\) e. \((5)^{-10}(2)^{10}\) f. \((-4)^{5}\left(\frac{1}{(16)^{12}}\right)\)

Verify that \(\left(a^{2}\right)^{3}=\left(a^{3}\right)^{2}\) using the rules of exponents.
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