Whole-number exponents Knowledge-Based Study

Study notes

1. What Is an Exponent?

   In mathematics, multiplication provides a concise way to write repeated addition. In a similar way, an exponent provides a compact notation for repeated multiplication of the same number.

   An exponential expression is written in the form:

   $$b^{n}$$

   In this expression:

   • Base ($b$): The factor that is being multiplied repeatedly.
   • Exponent ($n$): The number of times the base appears as a factor.

   For example, in the expression $3^{4}$, the base is 3 and the exponent is 4. This indicates that 3 is multiplied by itself 4 times:

   $$3^{4} = 3 \times 3 \times 3 \times 3 = 81$$

   Calculating Values of Powers

   When the exponent is a positive integer greater than 1, you evaluate the expression by expanding it into a product of identical factors and multiplying sequentially.

   Step-by-Step Examples

   • Evaluating $2^{5}$:
     Write 2 as a factor 5 times:
     $$2^{5} = 2 \times 2 \times 2 \times 2 \times 2$$
     Multiply pairwise: $2 \times 2 = 4$, $4 \times 2 = 8$, $8 \times 2 = 16$, and $16 \times 2 = 32$.
     Therefore, $2^{5} = 32$.
   • Evaluating $10^{3}$:
     Write 10 as a factor 3 times:
     $$10^{3} = 10 \times 10 \times 10 = 1{,}000$$
     Notice that for powers of 10 with a whole-number exponent $n$, the result is written as the digit 1 followed by exactly $n$ zeros.

   The Exponent 1

   Any non-zero number raised to the power of 1 is equal to the base itself:

   $$b^{1} = b$$

   For example, $7^{1} = 7$ and $125^{1} = 125$. Because a single factor of the number is present, its value remains unchanged.

   The Zero Exponent Rule

   For any non-zero base $b$, raising the base to the power of 0 yields 1:

   $$b^{0} = 1 \quad (b \neq 0)$$

   To see why this rule holds, observe the pattern of dividing by the base as the exponent decreases:

   • $2^{3} = 8$
   • $2^{2} = 8 \div 2 = 4$
   • $2^{1} = 4 \div 2 = 2$
   • $2^{0} = 2 \div 2 = 1$

   Continuing this decreasing division pattern demonstrates why $b^{0}$ must equal 1.

   Avoiding Common Errors

   A frequent error is multiplying the base by the exponent instead of evaluating repeated multiplication.

   • Incorrect: $4^{3} = 4 \times 3 = 12$
   • Correct: $4^{3} = 4 \times 4 \times 4 = 64$

   Summary of Rules for Whole-Number Exponents

   1. For $n \ge 1$, $b^{n} = \underbrace{b \times b \times \dots \times b}_{n\text{ factors}}$.
   2. For any base $b$, $b^{1} = b$.
   3. For any non-zero base $b$, $b^{0} = 1$.
   4. $1^{n} = 1$ for any whole number $n$.
   5. $0^{n} = 0$ for any whole number $n > 0$.