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Chapter 7: Problem 1

Write each expression with base 2 a) \(4^{6}\) b) \(8^{3}\) c) \(\left(\frac{1}{8}\right)^{2}\) d) 16

Short Answer

Expert verified
a) 2^{12}, b) 2^9, c) 2^{-6}, d) 2^4

Step by step solution

01

Understand the Powers of 2

Recognize that numbers like 4, 8, 16 can be written as powers of 2. Specifically: - 4 can be written as 2^2 - 8 can be written as 2^3 - 16 can be written as 2^4
02

Convert 4 to Base 2 in Expression a)

Since 4 = 2^2, substitute 4 with 2^2 in the expression: \[4^6 = (2^2)^6 = 2^{12}\]
03

Convert 8 to Base 2 in Expression b)

Since 8 = 2^3, substitute 8 with 2^3 in the expression: \[8^3 = (2^3)^3 = 2^9\]
04

Convert 8 to Base 2 in Expression c)

Since 8 = 2^3, substitute 8 with 2^3 in the fraction: \[\frac{1}{8} = 2^{-3}\] Then raise the result to the 2nd power: \[\bigg(\frac{1}{8}\bigg)^2 = (2^{-3})^2 = 2^{-6}\]
05

Express 16 in Base 2

Since 16 = 2^4, it directly converts to: \[16 = 2^4\]

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Key Concepts

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

Powers of 2
To understand how to write expressions in base 2, it鈥檚 crucial to know the concept of 'powers of 2'. This means expressing numbers as a product of 2 raised to an exponent. For example:
  • 4 can be written as \(2^2\)
  • 8 can be written as \(2^3\)
  • 16 can be written as \(2^4\)
Recognizing these patterns helps in converting numbers to base 2 more easily and is an essential skill for working with different bases in mathematics.
Exponential Expressions
An exponential expression involves raising a base number to a certain power. For instance, in the expression \(4^6\), 4 is the base and 6 is the exponent. Converting such expressions to base 2 involves a few steps.
First, identify how the base number can be rewritten as a power of 2. In the case of 4, since \(4 = 2^2\), the expression \(4^6\) can be transformed to \((2^2)^6\).
Using the laws of exponents, particularly the power of a power rule, simplify the expression to \(2^{12}\). This process is used for all given expressions to achieve their base 2 form.
Fractional Exponents
Sometimes, you need to work with fractional exponents or negative exponents, especially when dealing with fractions or roots. For example, consider the expression \(\bigg(\frac{1}{8}\bigg)^2\).
Notice that \(\frac{1}{8}\) can be written as \(8^{-1}\). Since 8 can also be written as \(2^3\), \(\frac{1}{8}\) becomes \(2^{-3}\).
Now, raising this to the power of 2, we get \((2^{-3})^2 = 2^{-6}\). Recognizing how to handle negative and fractional exponents allows you to convert a wider range of numbers to base 2 efficiently.
Understanding Mathematical Bases
A mathematical base refers to the number that gets multiplied by itself a certain number of times in an exponential expression. The most common base is 10, but base 2 is also frequently used, especially in computing and digital logic.
Converting numbers to different bases involves understanding how numbers can be decomposed into sums of powers of the new base. For example, 16 in base 2 can directly be written as \(2^4\).
This means that familiarity with different mathematical bases and how to express numbers in these bases is a crucial skill. Exercises that involve rewriting numbers in base 2 strengthen your ability to think in these different representations, making it easier to solve related problems in both academics and real-life applications like computer science.

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Most popular questions from this chapter

Scuba divers know that the deeper they dive, the more light is absorbed by the water above them. On a dive, Petra's light meter shows that the amount of light available decreases by \(10 \%\) for every \(10 \mathrm{m}\) that she descends. a) Write the exponential function that relates the amount, \(L,\) as a percent expressed as a decimal, of light available to the depth, \(d,\) in \(10-\mathrm{m}\) increments. b) Graph the function. c) What are the domain and range of the function for this situation? d) What percent of light will reach Petra if she dives to a depth of \(25 \mathrm{m} ?\)

The rate at which liquids cool can be modelled by an approximation of Newton's law of cooling, \(T(t)=\left(T_{i}-T_{f}\right)(0.9)^{\frac{t}{5}}+T_{f},\) where \(T_{f}\) represents the final temperature, in degrees Celsius; \(T_{i}\) represents the initial temperature, in degrees Celsius; and \(t\) represents the elapsed time, in minutes. Suppose a cup of coffee is at an initial temperature of \(95^{\circ} \mathrm{C}\) and cools to a temperature of \(20^{\circ} \mathrm{C}\). a) State the parameters \(a, b, h,\) and \(k\) for this situation. Describe the transformation that corresponds to each parameter. b) Sketch a graph showing the temperature of the coffee over a period of 200 min. c) What is the approximate temperature of the coffee after 100 min? d) What does the horizontal asymptote of the graph represent?

A biologist places agar, a gel made from seaweed, in a Petri dish and infects it with bacteria. She uses the measurement of the growth ring to estimate the number of bacteria present. The biologist finds that the bacteria increase in population at an exponential rate of \(20 \%\) every 2 days. a) If the culture starts with a population of 5000 bacteria, what is the transformed exponential function in the form \(P=a(c)^{b x}\) that represents the population, \(P,\) of the bacteria over time, \(x,\) in days? b) Describe the parameters used to create the transformed exponential function. c) Graph the transformed function and use it to predict the bacteria population after 9 days.

If a given population has a constant growth rate over time and is never limited by food or disease, it exhibits exponential growth. In this situation, the growth rate alone controls how quickly (or slowly) the population grows. If a population, \(P,\) of fish, in hundreds, experiences exponential growth at a rate of \(10 \%\) per year, it can be modelled by the exponential function \(P(t)=1.1^{t},\) where \(t\) is time, in years. a) Why is the base for the exponential function that models this situation \(1.1 ?\) b) Graph the function \(P(t)=1.1^{t} .\) What are the domain and range of the function? c) If the same population of fish decreased at a rate of \(5 \%\) per year, how would the base of the exponential model change? d) Graph the new function from part c). What are the domain and range of this function?

The Krumbein phi scale is used in geology to classify sediments such as silt, sand, and gravel by particle size. The scale is modelled by the function \(D(\varphi)=2^{-\varphi}\) where \(D\) is the diameter of the particle, in millimetres, and \(\varphi\) is the Krumbein scale value. Fine sand has a Krumbein scale value of approximately \(3 .\) Coarse gravel has a Krumbein scale value of approximately -5 a) Why would a coarse material have a negative scale value? b) How does the diameter of fine sand compare with the diameter of coarse gravel?
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