An exponent tells how many times a base is used as a factor, so \(2^4 = 2\times 2\times 2\times 2 = 16\). The order of operations, PEMDAS, says to evaluate parentheses first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.
Key facts:
- In \(b^n\), the base \(b\) is multiplied by itself \(n\) times , so \(3^4 = 81\), not \(12\).
- A power of 10 has as many zeros as its exponent: \(10^5 = 100,000\).
- PEMDAS order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
- An exponent affects only the base next to it: \(2\times 3^2 = 18\), but \((2\times 3)^2 = 36\).
What is an exponent?
An exponent is a small raised number that tells how many times a base is used as a factor. In \(5^3\), the base is \(5\) and the exponent is \(3\), so \(5^3 = 5\times 5\times 5 = 125\). Read \(5^2\) as “five squared” and \(5^3\) as “five cubed.”
A power is a number written as \(b^n\). The base \(b\) is the repeated factor and the exponent \(n\) counts the factors. For a whole number \(n \ge 2\), \(b^n\) means \(n\) copies of \(b\) multiplied together, and \(b^1 = b\).
An exponent is not repeated multiplication of the base by the exponent: \(2^4 = 2\times 2\times 2\times 2 = 16\), but \(2\times 4 = 8\).
What are powers of 10?
Powers of 10 follow a pattern: the exponent equals the number of zeros after the 1.
| Power | Product | Value |
|---|---|---|
| \(10^1\) | \(10\) | 10 |
| \(10^2\) | \(10\times 10\) | 100 |
| \(10^3\) | \(10\times 10\times 10\) | 1,000 |
| \(10^4\) | \(10\times 10\times 10\times 10\) | 10,000 |
| \(10^6\) | six factors of 10 | 1,000,000 |
What is the order of operations?
Without one agreed order, \(2 + 3\times 4\) could mean \(20\) or \(14\). Everyone uses the same order, remembered as PEMDAS:
- Parentheses and other grouping symbols, innermost first
- Exponents
- Multiplication and Division, from left to right
- Addition and Subtraction, from left to right
Multiplication and division share one level, and so do addition and subtraction. That is why \(24: 4\times 3 = 6\times 3 = 18\), not \(24: 12 = 2\).
How do you evaluate an expression with exponents and parentheses?
Work one operation at a time and rewrite the expression after each step. Example: evaluate \(4 + 3\times (8-2)^2: 6\).
- Parentheses: \(8-2 = 6\), so the expression is \(4 + 3\times 6^2: 6\).
- Exponents: \(6^2 = 36\), so it is \(4 + 3\times 36: 6\).
- Multiplication and division, left to right: \(3\times 36 = 108\), then \(108: 6 = 18\).
- Addition: \(4 + 18 = 22\).
An exponent applies only to the number or group directly to its left. In \(2\times 3^2\), only the 3 is squared: \(2\times 9 = 18\). With parentheses, \((2\times 3)^2 = 6^2 = 36\).
Exponents and Order of Operations: frequently asked questions
What does an exponent mean in math?
An exponent tells how many times to multiply the base by itself. For example, \(4^3\) means \(4\times 4\times 4 = 64\). The exponent is not multiplied by the base.
What does PEMDAS stand for?
PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. It lists the order in which operations are done. Multiplication and division share a level, as do addition and subtraction.
Do you always multiply before you divide?
No. Multiplication and division have equal rank, so work them from left to right as they appear. For example, \(36: 6\times 3 = 6\times 3 = 18\).
How do you write a large number as a power of 10?
Count the zeros after the 1. A number with 7 zeros after the 1, such as \(10,000,000\), equals \(10^7\).
What is the difference between 2 times 3 squared and 2 times 3, all squared?
Without parentheses, only the 3 is squared: \(2\times 3^2 = 2\times 9 = 18\). With parentheses, the whole product is squared: \((2\times 3)^2 = 6^2 = 36\).
See every chapter on the Grade 6 math page.