Powers Theory

Powers: Powers Theory

Powers Theory

Powers Rules
The following rules hold, assuming that # \displaystyle a > 0 # and # \displaystyle b > 0 #:# \displaystyle \begin{aligned} a^p \cdot a^q &= a^{p+q}, & \frac{1}{a^n} &= a^{-n}, & \frac{a^p}{a^q} &= a^{p-q}, & (a^p)^q &= a^{pq}, \\ (ab)^p &= a^p b^p, & \sqrt[n]{a} &= a^{\frac{1}{n}}, & \sqrt[q]{a^p} &= a^{\frac{p}{q}}, & \sqrt{ab} &= \sqrt{a}\sqrt{b}. \end{aligned} #

For example, using these rules we have:
# \displaystyle \frac{(3a^2b)^{\frac{1}{3}}}{2a^4b^{-\frac{1}{2}}} = \frac{3^{\frac{1}{3}}a^{\frac{2}{3}}b^{\frac{1}{3}}}{2a^4b^{-\frac{1}{2}}} = 3^{\frac{1}{3}} \cdot 2^{-1} a^{-\frac{10}{3}} b^{\frac{5}{6}}. #

The expression is reduced to a product of numbers and powers of the form
# \displaystyle C \cdot a^p b^q c^r \dots #

If # \displaystyle x \in \mathbb{R} # then
# \displaystyle \sqrt{x^2} = |x| \quad \text{and} \quad \sqrt[3]{x^3} = x, #
where # \displaystyle |x| # denotes the absolute value of # \displaystyle x #.

Absolute value

# \displaystyle |x| = \begin{cases} x & \text{if } x \geqslant 0, \\ -x & \text{if } x < 0. \end{cases} #

Note: # \displaystyle \sqrt{c^2} = |c| # but # \displaystyle x^2 = c^2 \Leftrightarrow x = c \lor x = -c. #

Powers (exponentials) and logarithms are inverses of each other. For instance:

# \displaystyle \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline x & \dots & -6 & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & \dots & \log_2 y \\ \hline 2^x & \dots & \frac{1}{64} & \frac{1}{32} & \frac{1}{16} & \frac{1}{8} & \frac{1}{4} & \frac{1}{2} & 1 & 2 & 4 & 8 & 16 & 32 & 64 & \dots & y \\ \hline \end{array} #