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28 Oct 2024
Madhavendra Thakur
October 28, 2024
Exponents allow us to represent repeated multiplication in a concise way. Mastering the basic exponent rules helps simplify complex algebraic expressions, and is a key stepping stone to more complex areas of math. This guide presents the rules with step-by-step examples for clarity.
When multiplying two expressions with the same base, add their exponents:
\[\begin{align*} a^m \cdot a^n = a^{m+n} \end{align*}\]
Example: Simplify \( x^3 \cdot x^4 \).
\[\begin{align*} x^3 \cdot x^4 &= (x \cdot x \cdot x) \cdot (x \cdot x \cdot x \cdot x) \\ &= x^7 \end{align*}\]
When dividing two expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator:
\[\begin{align*} \frac{a^m}{a^n} = a^{m-n} \quad \text{(where } a \neq 0\text{)} \end{align*}\]
Example: Simplify \( \dfrac{y^6}{y^2} \).
\[\begin{align*} \frac{y^6}{y^2} &= \frac{y \cdot y \cdot y \cdot y \cdot y \cdot y}{y \cdot y} \\ &= y^{6-2} = y^4 \quad \text{(cancel two } y\text{‘s from the top and bottom)} \end{align*}\]
When raising a power to another power, multiply the exponents:
\[\begin{align*} (a^m)^n = a^{m \cdot n} \end{align*}\]
Example: Simplify \( (x^2)^3 \).
\[\begin{align*} (x^2)^3 &= x^2 \cdot x^2 \cdot x^2 \\ &= x^{2+2+2} = x^6 \end{align*}\]
When raising a product to a power, apply the exponent to each part of the product:
\[\begin{align*} (ab)^n = a^n \cdot b^n \end{align*}\]
Example: Simplify \( (2x)^3 \).
\[\begin{align*} (2x)^3 &= (2 \cdot x) \cdot (2 \cdot x) \cdot (2 \cdot x) \\ &= 2^3 \cdot x^3 = 8x^3 \end{align*}\]
When raising a quotient to a power, apply the exponent to both the numerator and the denominator:
\[\begin{align*} \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad \text{(where } b \neq 0\text{)} \end{align*}\]
Example: Simplify \( \left(\frac{x}{3}\right)^2 \).
\[\begin{align*} \left(\frac{x}{3}\right)^2 &= \frac{x \cdot x}{3 \cdot 3} \\ &= \frac{x^2}{9} \end{align*}\]
Any non-zero base raised to the power of zero equals 1:
\[\begin{align*} a^0 = 1 \quad \text{(where } a \neq 0\text{)} \end{align*}\]
This can be derived using previous exponent rules:
\[\begin{align*} a^0 = a^{1 – 1} = \frac{a^1}{a^1} = \frac{a}{a} = 1. \end{align*}\]
Example: Simplify \( 5^0 \).
\[\begin{align*} 5^0 = 1. \end{align*}\]
A negative exponent means the reciprocal of the base raised to the positive exponent:
\[\begin{align*} a^{-n} = \frac{1}{a^n} \quad \text{(where } a \neq 0\text{)} \end{align*}\]
This can be derived using previous exponent rules. Recall that anything raised to the power of zero equals $1$, so:
\[\begin{align*} a^{-n} = a^{0 – n} = \frac{a^0}{a^n} = \frac{1}{a^n}. \end{align*}\]
Example: Simplify \( 3^{-2} \).
\[\begin{align*} 3^{-2} = \frac{1}{3^2} = \frac{1}{9}. \end{align*}\]
Example 1: Simplify \( \frac{(2x^3)^2 \cdot x^{-1}}{x^4} \).
\[\begin{align*} &\frac{(2x^3)^2 \cdot x^{-1}}{x^4} = \frac{2^2 \cdot x^{3 \cdot 2} \cdot x^{-1}}{x^4} \\ &= \frac{4 \cdot x^6 \cdot x^{-1}}{x^4} = \frac{4 \cdot x^{6 + (-1)}}{x^4} \\ &= \frac{4 \cdot x^5}{x^4} = 4x^{5-4} = 4x \end{align*}\]
Example 2: Simplify \( \frac{5x^{-2}}{10x^{-5}} \).
\[\begin{align*} &\frac{5x^{-2}}{10x^{-5}} = \frac{5}{10} \cdot \frac{x^{-2}}{x^{-5}} \\ &= \frac{1}{2} \cdot x^{(-2) – (-5)} \\ &= \frac{1}{2} \cdot x^{3} = \frac{x^3}{2} \end{align*}\]
Understanding and applying these exponent rules is crucial to developing a solid algebraic foundation. For more practice problems in math topics ranging from pre-algebra to differential equations, check out the I Aced Calculus app.
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