1.4.1: Rules of Exponents

1.4.2 Learning Objectives

• Interpret exponential expression with negative exponents
• Multiply exponential expressions with the same base using the Product Rule for Exponents
• Divide exponential expressions with the same base using the Quotient Rule for Exponents
• Use zero as an exponent

Negative Exponents

The first rule we wist to develop is the rule for interpreting negative exponents. A negative exponent implies that the exponential quantity is in the wrong position. For example:

\(b^{-1}=\dfrac{1}{b^{1}}=\dfrac{1}{b}\)

\(\dfrac{1}{b^{-1}}=b^{1}=b\)

Notice that the value of the base does not change.  The exponent changes to a positive value when it is moved into the correct position, being in the numerator or the denominator.

Example 1

Rewrite these exponential quantities with positive exponents.

1. \(2^{-1}\)
2. \(5^{-2}\)
3. \(\dfrac {1}{7^{-1}}\)
4. \(\dfrac {1}{3^{-4}}\)

Solution

1. \(2^{-1}=\dfrac {1}{2^{1}}=\dfrac {1}{2}\)
2. \(5^{-2}=\dfrac {1}{5^{2}}\)
3. \(\dfrac {1}{7^{-1}}=7^{1}=7\)
4. \(\dfrac {1}{3^{-4}}=3^{4}\)

The Product Rule for Exponents

The second rule we wish to develop is the rule for multiplying two exponential quantities having the same base and natural number exponents. The following examples suggest this rule:

\(
\begin{aligned}
&\begin{array}{c}
x^{2} \cdot x^{4}=\underbrace{x x}_{2} \cdot \underbrace{x x x x}_{4}=\underbrace{x x x x x x}_{6}=x^{6} \\
2+4=6
\end{array}\\
\end{aligned}
\)

\(
a \cdot a^{2}=\underbrace{a}_{1} \cdot \underbrace{aa}_{2}=\underbrace{a a a}_{3}=a^{3} \\
1 + 2 = 3
\)

Product Rule for Exponents

If \(x\) is a real number and \(n\) and \(m\) are natural numbers,

\(x^nx^m = x^{n+m}\)

To multiply two exponential quantities having the same base, add the exponents. Keep in mind that the exponential quantities being multiplied must have the same base for this rule to apply.

 

Example 2

Find the following products. All exponents are natural numbers.

1. \(2^3 \cdot 2^5 = 2^{3+5} = 2^8\)
2. \(a^6 \cdot a^14 = a^{6+14} = a^20\)
3. \(y^5 \cdot y = y^5 \cdot y^1 = y^{5+1} = y^6\)

Example 3

\(x^3y^4 \not = (xy)^{3+4}\)

Since the bases are not the same, the product rule does not apply.

The Quotient Rule for Exponents

The third rule we wish to develop is the rule for dividing two exponential quantities having the same base and natural number exponents.
The following examples suggest a rule for dividing two exponential quantities having the same base and natural number exponents.

\(
\dfrac{x^{5}}{x^{2}}=\dfrac{x x x x x}{x x}=\dfrac{\cancel{(x x)} x x x}{\cancel{(x x)}}=x x x=x^{3} . \text { Notice that } 5-2=3
\)

\(
\dfrac{a^8}{a^3} = \dfrac{aaaaaaaa}{aaa} = \dfrac{(\cancel{aaa})aaaaa}{(\cancel{aaa})} = aaaaa = a^5. \text { Notice that } 8-3=5
\)

Quotient Rule for Exponents

If \(x\) is a real number and \(n\) and \(m\) are natural numbers,

\(\dfrac{x^n}{x^m} = x^{n-m}, x \not = 0.\)

Example 3

Find the following quotients. All exponents are natural numbers.

1. \(\dfrac{3^5}{3^2} = 3^{5-2} = 3^3\)
2. \(\dfrac{27a^3b^6c^2}{3a^2bc} = 9a^{3-2}b^{6-1}c^{2-1} = 9ab^5c\)

When we make the subtraction, \(n-m\), in the division, \(\dfrac{x^n}{x^m}\), there are three possibilities for the values of the exponents:

The exponent of the numerator is greater than the exponent of the denominator, that is, \(n > m\). Thus, the exponent, \(n-m\), is a natural number.

The exponents are the same, that is, \(n = m\). Thus, the exponent, \(n - m\), is zero, a whole number.

The exponent of the denominator is greater than the exponent of the numerator, that is, \(n < m\). Thus, the exponent, \(n - m\), is a negative integer.

Zero as an Exponent

In the examples above, the exponents of the numerators were greater than the exponents of the denominators. Let’s study the case when the exponents are the same.

When the exponents are the same, say \(n\), the subtraction \(n-n\) produces \(0\).

Thus, by the second rule of exponents, \(\dfrac{x^n}{x^n} = x^{n-n} = x^0\)

But what real number, if any, does \(x^0\) represent? Let's think for a moment about our experience with division in arithmetic. We know that any nonzero number divided by itself is one.

\(\dfrac{8}{8} = 1\), \(\dfrac{43}{43} = 1\), \(\dfrac{258}{258} = 1\)

Since the letter \(x\) represents some nonzero real number divided by itself. Then \(\dfrac{x^n}{x^n} = 1\).

But we have also established that if \(x \not = 0\), \(\dfrac{x^n}{x^n} = x^0\). We now have that \(\dfrac{x^n}{x^n} = x^0\) and \(\dfrac{x^n}{x^n} = 1\). This implies that \(x^0 = 1\), \(x \not = 0\).

Exponents can now be natural numbers and zero. We have enlarged our collection of numbers that can be used as exponents from the collection of natural numbers to the collection of whole numbers.

Zero as an Exponent

If \(x \not = 0\), \(x^0 = 1\)

Any number, other than \(0\), raised to the power of \(0\), is \(1\). \(0^0\) has no meaning (it does not represent a number).

 

Example 4

Find each value. Assume the base is not zero.

1. \(6^0 = 1\)
2. \(246^0 = 1\)
3. \(4y^0 = 4 \cdot 1 = 4\)
4. \(\dfrac{y^6}{y^6} = y^0 = 1\)