Exponents and the laws of exponents

Repeated multiplication

Writing \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) gets old fast. Exponents give a shortcut: \(2^6\).

So \(2^6 = 64\), \(5^3 = 5 \cdot 5 \cdot 5 = 125\) and \(10^4 = 10{,}000\).

Exponents \(2\) and \(3\) have special names: \(7^2\) is "seven squared" and \(7^3\) is "seven cubed". The names come from geometry: a square with side \(7\) has area \(7^2\), and a cube with edge \(7\) has volume \(7^3\).

Powers of ten

Powers of \(10\) are easy: the exponent tells you how many zeros follow the \(1\).

Exponents of 1 and 0

Why would \(b^0\) be \(1\)? Look at the pattern for powers of \(3\):

Each time the exponent goes down by \(1\), the value is divided by \(3\). Following the pattern, \(3^0 = 3 \div 3 = 1\).

The laws of exponents

When powers share the same base, you can combine them without multiplying everything out.

Counting factors shows why the product rule works: \(5^2 \cdot 5^3 = (5 \cdot 5)(5 \cdot 5 \cdot 5)\), which is five factors of \(5\), or \(5^5\).

Exponents in longer expressions

In the order of operations, exponents come right after parentheses.

Practice