AI Math Beginner

An exponent tells you how many times to multiply the base by itself.

\[a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}\]

The base is the repeated factor (here 2) and the exponent is the count of multiplications (here 5), giving \(2^5 = 32\).

\[2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\]

Dividing \(a^n\) by itself gives 1, and the quotient rule gives \(a^n / a^n = a^{n-n} = a^0\), so \(a^0\) must equal 1.

\[a^0 = 1 \quad (a \neq 0)\]

\(x^1\) equals \(x\) itself, because raising to the first power means a single copy of the base.

\[x^1 = x\]

A negative exponent means take the reciprocal of the corresponding positive power.

\[x^{-n} = \frac{1}{x^n} \quad (x \neq 0)\]

Add the exponents and keep the base.

\[x^a \cdot x^b = x^{a+b}\]

Subtract the exponents and keep the base.

\[\frac{x^a}{x^b} = x^{a-b} \quad (x \neq 0)\]

When a power is raised to another power, multiply the exponents.

\[(x^a)^b = x^{a \cdot b}\]

An exponent applied to a product distributes to each factor in the product.

\[(xy)^n = x^n y^n\]

An exponent applied to a fraction distributes to both the numerator and the denominator.

\[\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n} \quad (y \neq 0)\]

\(x^{1/2}\) means the square root of \(x\), the non-negative number whose square is \(x\).

\[x^{1/2} = \sqrt{x}\]

\(x^{1/n}\) means the \(n\)th root of \(x\), the number which when raised to the \(n\)th power gives \(x\).

\[x^{1/n} = \sqrt[n]{x}\]