Arithmetic Reasoning Study Guide for the ASVAB

Page 4

Exponents and Radicals

These two concepts are actually opposite operations. An exponent tells us to multiply a number by itself a certain number of times. A radical asks us to find out what number, multiplied by itself, equals the number inside the radical sign. It’s similar to the relationship between multiplication and division, but a little different. Read on.

Exponents

An exponent tells us how many times to multiply a number by itself: Exponentiation is the act of using an exponent.

\(2^3\) tells us to multiply two by itself three times: \(2\cdot 2\cdot 2 =8\).

Exponents are also called powers. They can be positive or negative numbers. They can even be fractions, which will be discussed under “Radicals”.

A positive exponent on a number is the number of times a number is multiplied by itself.

\(8^5\) is the same as:

\[8 \cdot 8 \cdot 8 \cdot 8 \cdot 8\]

A negative exponent is the number of times 1 is divided by the number with the exponent.

\(8^{–5}\) is the same as:

\[\frac{1}{8 \cdot 8 \cdot 8 \cdot 8 \cdot 8}\]

So if you see an expression like this:

\[\frac{(a^4)(c^{–2})}{b^{–3}}\]

it is actually the same as:

\[\frac{(a^4)(b^3)}{c^2}\]

Rules for Operations with Exponents

There are three important rules to remember when simplifying and evaluating expressions having exponents:

For a number or variable multiplied by itself, the exponents can be added.

General rule 1: \(a^m\cdot a^n = a^{m+n}\)

Example: \(2^2\cdot 2^3 = 2^{2+3} = 2^5\)

For a number or variable divided by itself, the exponents can be subtracted.

General rule 2: \(a^m\div a^n = a^{m-n}\)

Example: \(3^5\div 3^2 = 3^{5-2} = 3^3\)

For a number or variable raised to a power, then to a power again, the exponents can be multiplied.

General rule 3: \({(a^m)}^n = a^{m\cdot n}\)

Example: \({(2^2)}^3 = 2^{2\cdot 3} = 2^6\)

Radicals

You may be familiar with roots, also called radicals, when they are presented inside the symbol \(\sqrt{}\), which goes by many names: radical sign, root symbol, radix, or surd. Recognize them, too, when they are presented as fractional exponents, as in:

\(x^{\frac{1}{2}}\), which is the same as \(\sqrt{x}\).

\[25^{\frac{1}{2}} = \sqrt{25} = 5\] \[y^{\frac{3}{2}} = (y^3)^{\frac{1}{2}} = \sqrt{y^3}\]

The word radicand is the name for whatever is inside the radical sign.

A radical is similar to an exponent but in the opposite direction; it tells us how many times some number is multiplied by itself to get the given number. A radical is written with this symbol \(\sqrt{}\) and a number in its small corner (if greater than 2), which is called the index. The most common radical is the square root of some number. This tells us we want to know what number multiplied by itself equals the given number.

A radical can also be written using a fraction as the power the base number is raised to. The denominator, like the index, tells us we want to know the number, which when multiplied by itself that number of times, equals the base number.

The square root of a number does not show an index because it is implicit: \(\sqrt{4}\). This is the same as using \(\frac{1}{2}\) as the power the base number is raised to.

\[\sqrt {4} = 4^{\frac{1}{2}} = 2\]

Logarithms

While logarithm questions do not commonly appear in the ASVAB pencil-and-paper test, they often do in CAT-ASVAB. Students often eye logarithms suspiciously. Well, you shouldn’t. It’s simply a variation of exponents.

The expression \(log_2 (8)\) simply asks the question:

“How many times will you have to multiply 2 (the “base”) by itself to get 8?” The answer is three.

Therefore, \(log_2 (8) = 3\) (Read as: “The log base 2 of 8 is 3.”)

Its exponential form is: \(2^3 = 8\)

Logarithmic expressions often appear in equations where you must solve the value of a variable. Consider the following question:

Solve for x if:

\[log_3(2x + 1) = log_3(9x)\]

The logarithmic expressions on the left and right sides of the equation have the same base (3), and the only way for the expressions to be equal is for the arguments, or the terms inside the parentheses, to be equal. Thus, you can proceed to solve with:

\[2x + 1 = 9x\]

which will give you the value of:

\[x = \frac{1}{7}\]

Scientific Notation

A number written in scientific notation is expressed as some decimal number whose absolute value is greater than 1 and less than 10, times some power of 10. Writing numbers in this form is useful in describing and talking about very large or very small numbers. A number like 13,549,000,000 is hard to say and harder to understand, but scientific notation makes it easy by putting the magnitude of that number in the power of ten. \(1.3549\cdot 10^{10}\) is easier to grasp than the long string of numbers.

Factorials

A factorial is made using the exclamation point symbol (!). It tells us to multiply a counting number by decreasing increments from itself in whole units, starting with one less than the original number, until the number one is reached.

For example, 6!, which is spoken and read as “six factorial”, is:

\(6\cdot 5\cdot 4\cdot 3\cdot 2 \cdot 1\)
\(30\cdot 4\cdot 3\cdot 2\cdot 1\)
\(120\cdot 3\cdot 2\cdot 1\)
\(360\cdot 2\cdot 1\)
\(720\)

Six is multiplied by descending increments of whole counting numbers, starting with one less than itself.

Measurement

The two most commonly used measurement systems are the metric system, which is used in most of the world, and the system used in the U.S., known officially as the U.S. Customary system. Even in the U.S., you are very likely to run into metric units, as many products we use here are made in other countries, and some American manufacturers use metric parts.

The U.S. Customary System

  • Length—The basic unit of length is the inch. Here are some others:

    • 1 foot = 12 inches
    • 1 yard = 36 inches or 3 feet
    • 1 mile = 1,760 yards or 5,280 feet

  • Volume—The basic unit of volume is the fluid ounce. Here are some others:

    • 1 cup = 8 fluid ounces
    • 1 pint = 16 fluid ounces or 2 cups
    • 1 quart = 32 fluid ounces or 2 pints
    • 1 gallon = 128 fluid ounces or 4 quarts

  • Weight—The basic unit of weight is the ounce. Here are some others:

    • 1 pound = 16 ounces
    • 1 ton = 2000 pounds

The Metric System

First, the metric system uses a number of prefixes. Below are the most commonly used ones and what they mean. Think of each prefix as a multiplier.

kilo- — times 1000
hecto- — times 100
deka- — times 10
deci- — times 0.1
centi- — times 0.01
milli- — times 0.001

  • Length—The basic unit of length is the meter, abbreviated m. Here are some others:

    • 1 kilometer (km) = 1000 m
    • 1 dekameter (dam) = 10 m
    • 1 centimeter (cm) = 0.01 m
    • 1 millimeter (mm) = 0.001 m

  • Volume—The basic unit of volume is the liter, abbreviated L. Here are some others:

    • 1 kiloliter (kL) = 1000 L
    • 1 centiliter (cL) = 0.01 L
    • 1 milliliter (mL) = 0.001 L
    • 1 cubic centimeter (cc or \(\text{cm}^3\)) = 1 mL

  • Weight—The basic unit of weight (technically mass) is the gram,abbreviated g. Here are some others:

    • 1 kilogram (kg) = 1000 g
    • 1 centigram (cg) = 0.01 g
    • 1 milligram (mg) = 0.001 g

Note: technically, the base unit of mass is the kilogram, but that won’t be an issue on this test.

Converting Between the Two Systems

You probably won’t have to know any of this for the test, but here are some really basic metric/standard conversions that may help you understand the two systems better.

1 inch = 2.54 centimeters = 25.4 millimeters
1 foot = 30.5 centimeters
3.28 feet = 1 meter = 39.37 inches
1 cup = 0.236 liter = 237 milliliter
1 quart = 0.946 liter = 946 milliliters
1 lb = 454 g
2.2 lb = 1 kilogram = 1000 g

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