Integer exponent rules Knowledge-Based Study

Study notes

1. What Is an Exponent?

   An exponential expression consists of a base and an exponent (or power). When the exponent is a positive integer, it indicates how many times the base is multiplied by itself.

   For any non-zero real number $a$ and positive integer $n$:

   $$a^n = \underbrace{a \times a \times \dots \times a}_{n\text{ factors}}$$

   Zero Exponent Rule

   Any non-zero base raised to the power of zero equals 1:

   $$a^0 = 1 \quad (a \neq 0)$$

   Note: The expression $0^0$ is indeterminate and undefined in standard arithmetic.

   Negative Exponent Rule

   A negative exponent indicates the reciprocal of the base raised to the positive power:

   $$a^{-n} = \frac{1}{a^n} \quad \text{and} \quad \frac{1}{a^{-n}} = a^n \quad (a \neq 0)$$

   For example: $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

   Operations with the Same Base

   When multiplying or dividing powers that share the same base, specific rules simplify the expressions without having to compute large numbers directly.

   The Product Rule

   To multiply powers with the same base, keep the base and add the exponents:

   $$a^m \times a^n = a^{m+n}$$

   Example:

   • $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$
   • $5^4 \times 5^{-6} = 5^{4 + (-6)} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

   The Quotient Rule

   To divide powers with the same base, keep the base and subtract the exponent of the denominator from the exponent of the numerator:

   $$\frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)$$

   Example:

   • $\frac{7^6}{7^2} = 7^{6-2} = 7^4$
   • $\frac{4^2}{4^5} = 4^{2-5} = 4^{-3} = \frac{1}{4^3} = \frac{1}{64}$

   Raising Powers to Powers

   Further rules apply when an exponential term or product is itself raised to an exponent.

   Power of a Power Rule

   To raise a power to another power, keep the base and multiply the exponents:

   $$(a^m)^n = a^{m \cdot n}$$

   Example: $(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$

   Power of a Product Rule

   To raise a product to a power, raise each factor to the power:

   $$(ab)^n = a^n b^n$$

   Example: $(2x)^3 = 2^3 \cdot x^3 = 8x^3$

   Power of a Quotient Rule

   To raise a fraction to a power, raise both the numerator and the denominator to that power:

   $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)$$

   When the exponent is negative, invert the fraction and change the sign of the exponent:

   $$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n}$$

   Simplifying Complex Expressions

   When working with expressions involving multiple rules, apply operations methodically, following standard order of operations.

   Example 1

   Simplify the expression so that it contains only positive exponents:

   $$\frac{(x^3 y^{-2})^2}{x^{-4} y^3}$$

   Step-by-step simplification:

   1. Apply the power of a product rule to the numerator: $(x^3 y^{-2})^2 = x^{3 \cdot 2} y^{-2 \cdot 2} = x^6 y^{-4}$.
   2. Write the full fraction: $\frac{x^6 y^{-4}}{x^{-4} y^3}$.
   3. Apply the quotient rule to like bases:
      • For base $x$: $x^{6 - (-4)} = x^{6 + 4} = x^{10}$
      • For base $y$: $y^{-4 - 3} = y^{-7}$
   4. Rewrite with positive exponents: $x^{10} y^{-7} = \frac{x^{10}}{y^7}$.

   Summary of Key Rules

   • $a^0 = 1$
   • $a^{-n} = \frac{1}{a^n}$
   • $a^m \cdot a^n = a^{m+n}$
   • $\frac{a^m}{a^n} = a^{m-n}$
   • $(a^m)^n = a^{m \cdot n}$
   • $(ab)^n = a^n b^n$