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Chapter 4: Problem 5

Evaluate and write the result using scientific notation: a. \(\left(2.3 \cdot 10^{4}\right)\left(2.0 \cdot 10^{6}\right)\) b. \(\left(3.7 \cdot 10^{-5}\right)\left(1.1 \cdot 10^{8}\right)\) c. \(\frac{8.19 \cdot 10^{23}}{5.37 \cdot 10^{12}}\) d. \(\frac{3.25 \cdot 10^{8}}{6.29 \cdot 10^{15}}\) e. \(\left(6.2 \cdot 10^{52}\right)^{3}\) f. \(\left(5.1 \cdot 10^{-11}\right)^{2}\)

Short Answer

Expert verified
a) 4.6 \cdot 10^{10}\) b) 4.07 \cdot 10^{3}\ c) 1.524 \cdot 10^{11}\ d) 5.16 \cdot 10^{-8}\ e) 2.38328 \cdot 10^{158}\ f) 2.601 \cdot 10^{-21}\

Step by step solution

01

Multiply the bases for part (a)

Calculate the product of the base numbers: 2.3 and 2.0. ewline\(2.3 \times 2.0 = 4.6\)
02

Add the exponents for part (a)

Add the exponents of the powers of 10: ewline\(10^{4} + 10^{6} = 10^{(4+6)} = 10^{10}\)Compute: \(\left(2.3 \cdot 10^{4}\right) \left(2.0 \cdot 10^{6}\right) = 4.6 \cdot 10^{10}\)
03

Multiply the bases for part (b)

Calculate the product of the base numbers: 3.7 and 1.1. ewline\(3.7 \times 1.1 = 4.07\)
04

Add the exponents for part (b)

Add the exponents of the powers of 10: ewline\(10^{-5} + 10^{8} = 10^{(-5+8)} = 10^{3}\)Compute: \(\left(3.7 \cdot 10^{-5}\right) \left(1.1 \cdot 10^{8}\right) = 4.07 \cdot 10^{3}\)
05

Divide the bases for part (c)

Calculate the quotient of the base numbers: 8.19 and 5.37. \(8.19 \div 5.37 \approx 1.524\)
06

Subtract the exponents for part (c)

Subtract the exponents of the powers of 10: ewline\(10^{23} - 10^{12} = 10^{(23-12)} = 10^{11}\)Compute: \(\frac{8.19 \cdot 10^{23}}{5.37 \cdot 10^{12}} \approx 1.524 \cdot 10^{11} \)
07

Divide the bases for part (d)

Calculate the quotient of the base numbers: 3.25 and 6.29. ewline\(3.25 \div 6.29 \approx 0.516\)
08

Subtract the exponents for part (d)

Subtract the exponents of the powers of 10: ewline\(10^{8} - 10^{15} = 10^{(8-15)} = 10^{-7}\)Compute: \(\frac{3.25 \cdot 10^{8}}{6.29 \cdot 10^{15}} \approx 0.516 \cdot 10^{-7} \rightarrow 5.16 \cdot 10^{-8}\)
09

Raise the base to the power for part (e)

Raise the base 6.2 to the power of 3. ewline\(6.2^{3} = 238.328\)
10

Multiply the exponents for part (e)

Multiply the exponent 52 by 3. ewline\((10^{52})^{3} = 10^{(52 \cdot 3)} = 10^{156}\)Compute: \((6.2 \cdot 10^{52})^{3} = 238.328 \cdot 10^{156}\rightarrow2.38328 \cdot 10^{158} \)
11

Square the base for part (f)

Square the base 5.1. ewline\(5.1^{2} = 26.01\)
12

Multiply the exponents for part (f)

Multiply the exponent -11 by 2. ewline\((10^{-11})^{2} = 10^{(-11 \cdot 2)} = 10^{-22}\)Compute: \((5.1 \cdot 10^{-11})^{2} = 26.01 \cdot 10^{-22}\rightarrow 2.601 \cdot 10^{-21}\)

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Key Concepts

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

Multiplication of Exponents
When multiplying numbers in scientific notation, you need to multiply the base numbers and add the exponents.
Simply follow these steps:
1. Multiply the base numbers.
For example, given \( \left( 2.3 \cdot 10^{4} \right) \left( 2.0 \cdot 10^6 \right) \): Multiply 2.3 by 2.0 to get 4.6.
2. Add the exponents.
In our example, add \( 4 \) and \( 6 \) to get \( 10^{10} \).
3. Combine your results to get the final answer: \( 4.6 \cdot 10^{10} \).
This method also applies when dealing with negative exponents or zero.
Division of Exponents
To divide numbers in scientific notation, you divide the base numbers and subtract the exponents.
Here鈥檚 how it's done:
1. Divide the base numbers.
For example, given \( \frac{8.19 \cdot 10^{23}}{5.37 \cdot 10^{12}} \): Divide 8.19 by 5.37 to get approximately 1.524.
2. Subtract the exponents.
For our example, subtract \( 12 \) from \( 23 \) to get \( 10^{11} \).
3. Combine your results: \( 1.524 \cdot 10^{11} \).
Follow this same process with negative or smaller exponents as well.
Exponent Rules
Understanding exponent rules is crucial for working with scientific notation.
Here are some key rules:
  • When multiplying, add the exponents: \[ a^m \times a^n = a^{m+n} \]
  • When dividing, subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n} \]
  • Raising an exponent to another power, multiply the exponents: \[ (a^m)^n = a^{m\times n} \]
Let鈥檚 apply these rules:
For multiplication: \( \left( 3.7 \cdot 10^{-5} \right) \left( 1.1 \cdot 10^8 \right) \rightarrow 3.7 \cdot 1.1 = 4.07 \) and sum of exponents: \( -5 + 8 = 3 \) results in \( 4.07 \cdot 10^3 \).
For division: \( \frac{3.25 \cdot 10^8}{6.29 \cdot 10^{15}} \rightarrow 3.25 \div 6.29 \approxeq 0.516 \) and subtracting the exponents: \( 8 - 15 = -7 \) which simplifies to \( 5.16 \cdot 10^{-8} \).
These critical rules ease the process of working with large or small numbers.
Base Number Operations
Base number operations in scientific notation often involve straightforward multiplication, division, and exponentiation.
Here are the key steps:
  • Multiplication: Multiply the base numbers directly.
  • Division: Divide the base numbers directly.
  • Exponentiation: Raise the base number to the given power.
For example:
Given \( \( 6.2 \cdot 10^{52} \)^3 \rightarrow 6.2^{3} = 238.328 \) and multiply the exponent: \( 52 \cdot 3 = 156 \), the result is \( 238.328 \cdot 10^{156} \).
Another example:
Square \( \left( 5.1 \cdot 10^{-11} \right)^2 \rightarrow 5.1^2 = 26.01 \) and multiply the exponent: \( -11 \cdot 2 = -22 \), resulting in \( 26.01 \cdot 10^{-22} \).
These step-by-step operations help simplify complicated calculations.

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Most popular questions from this chapter

In 2006 the United Kingdom generated approximately 81 terawatt-hours of nuclear energy for a population of about 60.6 million on 94,525 miles \(^{2}\). In the same year the United States generated approximately 780 terawatt-hours of nuclear energy for a population of about 300 million on 3,675,031 miles \(^{2}\). A terawatt is \(10^{12}\) watts. a. How many terawatt-hours is the United Kingdom generating per person? How many terawatt-hours is it generating per square mile? Express each in scientific notation. b. How many terawatt-hours is the United States generating per person? How many terawatt-hours are we generating per square mile? Express each in scientific notation. c. How much nuclear energy is being generated in the United Kingdom per square mile relative to the United States? d. Write a brief statement comparing the relative magnitude of generation of nuclear power per person in the United Kingdom and the United States.

Using rules of exponents, show that \(\frac{1}{x^{-n}}=x^{n}\).

Simplify each expression using two different methods, and then compare your answers. Method I: Simplify inside the parentheses first, and then apply the exponent rule outside the parentheses. Method II: Apply the exponent rule outside the parentheses, and then simplify. a. \(\left(\frac{m^{2} n^{3}}{m n}\right)^{2}\) b. \(\left(\frac{2 a^{2} b^{3}}{a b^{2}}\right)^{4}\)

How many droplets of water are in a river that is \(100 \mathrm{~km}\) long, \(250 \mathrm{~m}\) wide, and \(25 \mathrm{~m}\) deep? Assume a droplet is 1 milliliter. ( Note: one liter \(=\) one cubic decimeter and 10 decimeters \(=\) I meter.)

(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. \((-4)^{5}\left(\frac{1}{(16)^{12}}\right)\)
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