exponents

Exponents

Exponents tell how many times to use a number itself in multiplication. There are different laws that guide calculations involving exponents. In this chapter, we will explore how these laws are used. For example: $4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^6$ and $5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^7$; we have multiplied alike factors and got answers which are $4^6$ and $5^7$. 4 and 5 are our factors and they are called bases while 6 and 7 are called exponents. $4^6$ is read as 'sixth power of four' or 'four to the sixth power', and $5^7$ is read as 'seventh power of five' or 'five to the seventh power'. $4 \times 4 \times 4 \times 4 \times 4 \times 4$ is the expanded form of $4^6$ and the expanded form of $5^7$ is $5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5$. The product of any expanded form is called a power of the factor. $4^6$ and $5^7$ are the powers of their respective factors. $4^6$ means multiply 4 six times and $5^7$ means multiply 5 seven times. Indication of power, base, and exponent is done as follows: Base: The number being raised to a power (e.g., 4 or 5). Exponent: The number that tells how many times the base is multiplied by itself (e.g., 6 or 7). Power: The expression of base raised to an exponent (e.g., $4^6$, $5^7$).

The Laws of Exponents

First law: Multiplication of positive integral exponents When multiplying powers having the same base, we add their exponents: $x^m \cdot x^n = x^{m+n}$, where $x$ is any base and $m, n$ are integers. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$. Second law: Division of positive integral exponents When dividing powers of the same base, we subtract the exponents (subtract the exponent of the divisor from the exponent of the dividend): $\frac{x^m}{x^n} = x^{m-n}$, where $x$ is a real number and $m, n$ are integers. For example, $\frac{3^5}{3^2} = 3^{5-2} = 3^3$. Third law: Zero exponents Any non-zero number raised to the power of zero is equal to 1: $x^0 = 1$, where $x$ is any real number except zero. Note that $0^0$ is undefined. For example, $5^0 = 1$. Fourth law: Negative exponents A negative exponent means taking the reciprocal of the base and raising it to the opposite positive exponent: $x^{-n} = \frac{1}{x^n}$, where $x \neq 0$ and $n$ is an integer. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Verification of the Laws of Exponents

First law (Multiplication of powers with the same base): For example, if $x = 3$, $m = 2$, and $n = 4$, then: $3^2 \cdot 3^4 = 3^{2+4} = 3^6$. Second law (Division of powers with the same base): For example, if $x = 5$, $m = 6$, and $n = 2$, then: $\frac{5^6}{5^2} = 5^{6-2} = 5^4$. Third law (Zero exponents): For example, if $x = 8$, then: $8^0 = 1$. Fourth law (Negative exponents): For example, if $x = 10$ and $n = 3$, then: $10^{-3} = \frac{1}{10^3} = \frac{1}{1000}$.

Examples of Exponent Rules

Example 1: Simplify $(x^3)^4$. Using the first law, $(x^3)^4 = x^{3 \times 4} = x^{12}$. Example 2: Simplify $\frac{x^5}{x^2}$. Using the second law, $\frac{x^5}{x^2} = x^{5-2} = x^3$. Example 3: Simplify $3^{-2}$. Using the fourth law, $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.