Tuesday, August 19, 2025

Algebra of real exponents

Algebra of real exponents

Let’s break down the algebra of real exponents in a clear, structured way.


1. Basic Meaning

For any real number exponent $r$ and positive real base $a > 0$:

$$ a^r \quad \text{means “a multiplied by itself r times” (extended to fractional and negative r)}. $$

  • If $r$ is an integer: $a^n = a \cdot a \cdots a$ (n times).
  • If $r = \frac{p}{q}$ (fraction): $a^{p/q} = \sqrt[q]{a^p}$.
  • If $r$ is negative: $a^{-r} = \frac{1}{a^r}$.
  • If $r$ is any real number, it is defined using limits or logarithms: $a^r = e^{r \ln a}$.

2. Exponent Laws (Valid for $a > 0$)

  1. Product Rule

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

  1. Quotient Rule

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

  1. Power of a Power

$$ (a^m)^n = a^{mn} $$

  1. Power of a Product

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

  1. Power of a Quotient

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


3. Special Exponents

  • $a^0 = 1$ (for $a \neq 0$)
  • $a^1 = a$
  • $a^{-n} = \frac{1}{a^n}$
  • $a^{1/n} = \sqrt[n]{a}$ (n-th root)

4. Examples

  1. $2^{3/2} = \sqrt{2^3} = \sqrt{8}$
  2. $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
  3. $(9^{1/2})^4 = 3^4 = 81$
  4. $2^3 \cdot 2^{1.5} = 2^{4.5} = 2^{9/2} = \sqrt{512}$

5. Common Pitfalls

  • Negative bases with fractional exponents can be undefined in the reals (e.g., $(-8)^{1/3}$ is fine, but $(-8)^{1/2}$ is not a real number).
  • The rules require same base unless you rewrite using logs.
  • $a^{m+n} \neq a^m + a^n$ — addition does not work that way.

If you’d like, I can make you a visual “cheat sheet” diagram showing all real exponent rules and examples on one page. Would you like me to create that?

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