Next Generation Quantitative Reasoning, Algebra, and Statistics Study Guide for the ACCUPLACER Test
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Exponents
Just as multiplication enables many additions to be performed simultaneously—rather than writing \(2 + 2 + 2 + 2 + 2\), we write \(2 \cdot 5\)—many multiplications can be expressed with exponents. For example, \(2 \cdot 2 \cdot 2\) can be expressed as \(2^3\).
Basic Exponents
Consider the following expression: \(2^2\), which is read “two squared” or “two to the second power.”
It is equivalent to \(2 \cdot 2\), or \(4\).
The expression \(2^3\) is read “two cubed” or “two to the third power” and is equal to \(2 \cdot 2 \cdot 2\) or \(8\).
The expression \(x^y\) is read “\(x\) to the \(y\) power.”
Radicals
Fractional exponents that contain a denominator larger than \(1\) can be written in radical form. For example, the expression \(4^{\frac{1}{2}}\) (read “four to the one-half power”) is equivalent to \(\sqrt{4}\).
Likewise, expressions containing a fractional exponent with a denominator of \(3\) can be written as a cube root:
\[x^{\frac{1}{3}} = \sqrt[3]{x}\]Notice that both \(2^2=4\) and \((-2)^2=4\) — which do we use when taking the square root? The same question arises for numbers raised to the powers of \(\frac{1}{4}\), \(\frac{1}{6}\), \(\frac{1}{8}\), and so on, that is, for any fractional exponent which has an even denominator when the fraction is written in reduced form. In general, we use the square root symbol to denote the positive root, which we call the principal root. Continuing the previous example, \(\sqrt{4}=2\) is the principal root. If we want to solve an equation with it, however, we must remember to include both the positive and negative solutions. For example, when solving for \(x\) in \((x+1)^2=9\) we should remember both the positive and negative roots. This is often written with a “plus-minus” sign \(\pm\). In this example, \(x+1=\pm\sqrt{9}\) so \(x=2\) or \(x=-4\).
Fractional Exponents
Fractional exponents may contain an integer greater than \(1\) in both the numerator and the denominator. Consider the following examples:
\[2^{\frac{5}{3}}\] \[3^{\frac{2}{3}}\] \[5^{\frac{3}{2}}\]To evaluate fractional exponents, raise the base to the power of the numerator, and then evaluate the \(n\)th root of the resulting value, where \(n\)th is the number in the denominator.
Evaluate this expression:
\[4^{\frac{3}{2}}\]Begin by raising the base to the third power.
Then evaluate the square root of the result:
Scientific Notation
Scientific notation enables very large or very small numbers to be condensed and expressed as decimals multiplied by \(\bf{10}\) raised to an exponent. Many scientific measurements and calculations involve very large and very small numbers, and the ability to express these numbers without writing dozens of digits avoids tediousness.
Consider this number:
\[1,204,000,000,000,000\]which can be expressed in scientific notation as:
\[1.204 \cdot 10^{15}\]When \(1.204\) is multiplied by \(10^{15}\), the decimal place is moved \(15\) places to the right, yielding the original number.
Likewise, \(0.0000000002203\) can be expressed as \(2.203 \cdot 10^{-10}\).
When \(2.203\) is multiplied by \(10^{-10}\), the decimal place moves \(10\) places to the left, yielding the original number.
When performing calculations involving scientific notation with your calculator, represent each multiplication portion inside of parentheses to avoid any potential order of operations errors.
For example, if dividing \(2.2 \cdot 10^{-23}\) by \(1.9 \cdot 10^{7}\), into the calculator you would enter:
Operations with Exponents
Familiarize yourself with the following rules of exponents and understand their derivation.
\[x^0 = 1 \;\text{when}\; x \ne 0\] \[x^a \cdot x^b = x^{a + b}\] \[x^a \div x^b = x^{a - b}\] \[(x^a)^b = x^{a \cdot b}\] \[x^{-a} = \frac{1}{x^a}\]When multiplying two exponential terms, only exponential terms that share the same base can be combined without first evaluating the exponent. With whole numbers of differing bases, we can always evaluate the exponents without a calculator, and then we can multiply. For example:
\[2^3 \cdot 3^2 = 8 \cdot 9 = 72\]Consider this problem: Simplify the expression:
\[\left( \frac{x^3 \cdot x^2}{x^{-5}} \right) ^2\]Begin by eliminating the negative exponent:
\[(x^3 \cdot x^2 \cdot x^5)^2\]Add the exponents inside of the parentheses:
\[(x^{3 + 2 + 5})^2 = (x^{10})^2\]Finally, multiply the exponents:
\[(x^{10})^2 = x^{10 \cdot 2} = x^{20}\]Algebraic Expressions
Algebraic expressions do not contain an equal sign; this is significant because expressions cannot be solved, they can only be reduced or simplified. Reducing algebraic expressions entails proper use of the order of operations and variables (if the expression involves unknown quantities).
Create an Expression
The ability to correctly translate word problems into algebraic expressions and equations is crucial for success in STEM fields. Creating expressions entails assigning variables to unknown values and relating these variables to known values through mathematical operations. If only a single value is unknown, but the correct relationship between the values is produced, the unknown quantity can be found. In the case of two unknowns, two equations must be produced before each variable can be computed.
Consider the following example: The average of four consecutive odd numbers is \(68\). What is the sum of the smallest and largest number?
To solve: Use \(x\) to represent the smallest odd number. Each of the following odd numbers can be found by adding \(2\) to the preceding number. The average of the values is their sum divided by the total number of numbers, and the average is given, so an equation can be written:
\[\frac{x + (x + 2) + (x + 4) + (x + 6)}{4} = 68\]Multiply both sides by \(4\) and combine like terms:
\[4x + 12 = 272\] \[x = 65, \;\text{where}\;x \;\text{is the smallest number}\]Recall that the question is not asking for the smallest number, but the sum of the smallest and largest numbers:
\[65 + (65 + 6) = 136\]Evaluate an Expression
Evaluating an expression entails simplifying a given expression by performing algebraic operations. Sometimes this involves replacing a variable with its given value and computing a quantity.
Properties of Operations
There are a few algebraic properties with which you should be familiar.
The associative property for addition is described algebraically as:
\[(x + y) + z = x + (y + z)\]The associative property for multiplication is described as:
\[(x \cdot y) \cdot z = x \cdot (y \cdot z)\]The commutative property of addition:
\[x + y = y + x\]The commutative property of multiplication:
\[x \cdot y = y \cdot x\]The distributive property:
\[x (y + z) = xy + xz\]Of these, it is most important to be able to use the distributive property when evaluating expressions. As an example, evaluate the following expression:
\[3(x + 2) - 2(x + y) + 2(-3x - y)\]Begin by distributing each coefficient through the parentheses:
\[3x + 6 - 2x - 2y - 6x - 2y\]Combine like terms:
\[-5x - 4y + 6\]Combining Like Terms
Like terms are those that are equivalent in both type and degree. (The degree of an exponent is just another way of saying the power it’s raised to. For example, \(5^4\) could also be read as “five to the fourth degree”.) Consider this expression:
\[2 - 3x - x + y + 2x^2 - 5xy - x^2\]which simplifies to \(2 - 4x + y + x^2 - 5xy\), after combining like terms.
The \(2\) cannot combine with any of the other terms because each of the other terms contains a variable.
The terms containing an \(x\) to the first power can be combined \(-3x - x = -4x\), but these terms cannot be combined with those containing an \(x\) to the second power.
Although the \(-5xy\) term contains both an \(x\) and a \(y\) raised to the first power, it cannot be combined with terms containing only one of \(x\) or \(y\).
Determining if Expressions are Equal
In order for two expressions to be equal, they must evaluate to the same value. Consider this example:
If \(x = 2\), determine what \(y\) must equal in order for the following equation to be true:
\[4(x + 3) - 2 = -3y\]Begin by substituting the value of \(x\) into the expression on the left (do this before distributing the \(4\) in order to save yourself some calculation steps):
\[4(2 + 3) - 2 = -3y\]which becomes
\[4(5) - 2 = -3y\]which becomes
\[18 = -3y\]Divide both sides by \(-3\) to find the value of \(y\) that makes the equation true,
\[-6 = y\]All Study Guides for the ACCUPLACER Test are now available as downloadable PDFs