Laws of Exponents with Integer Exponents
Key points
- Exponents are expressions used to represent repeated multiplication of numbers and/or variables.
- Laws of exponents are a set of rules used to perform a variety of algebraic operations with exponents and simplify expressions.
- These include: the multiplication rule, the division rule, the power of a power/product/fraction rule, the zero and negative exponent rules, and the fractional exponent rule.
Exponents are mathematical expressions used to represent repeated multiplication of either integers or variables.
- The structure of an exponent is composed of:
- A base, which is the number or variable that is being multiplied repeatedly.
- An exponent, which is written as a superscript to the right of the base. It represents the number of times that the base is being multiplied by itself.
- In the example to the right, 5 is the base, as 3 is the exponent.
Rules or laws of exponents are used to simplify exponential expressions. They can be used with numbers or variables. They are used when exponents have the same base. There are 8 main exponent rules:
- The multiplication/product rule – to multiply powers, keep the base and add the exponents.
- The division/quotient rule – to divide powers, keep the base and subtract the exponents.
- The power of a power rule – when an exponential expression is lifted to another power, keep the base, and multiply the exponents.
- The power of a product rule – exponents apply to all terms within a set of parentheses.
- Power of a fraction rule – an exponent outside of a fraction is distributed to each number or variable.
- Zero exponent rule – anything elevated to the power of 0 has a value of 1.
- The negative exponent rule – if there is a negative exponent, set it as the reciprocal and drop the negative.
- The fractional exponent rule – the denominator becomes the root power, and the numerator is carried into the radical expression.
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