6.1: Properties of Exponents

The Basic Properties of Exponents.

In the realm of algebra, exponents serve as fundamental tools that simplify and manipulate expressions involving repeated multiplication. Mastering the rules of exponents is crucial for effectively solving equations and simplifying expressions.

Here, we will cover 7 fundamental rules for exponents. If you don't know these well, you won't do well in practice problems!

1. Product Rule

The product rule states that when multiplying two exponential expressions with the same base, you can add the exponents:

$a^m \times a^n = a^{m + n}$

For instance, $2^3 \times 2^4 = 2^{3 + 4} = 2^7$.

2. Quotient Rule

The quotient rule dictates that when dividing two exponential expressions with the same base, you can subtract the exponent of the divisor from the exponent of the dividend:

$\frac{{a^m}}{{a^n}} = a^{m - n}$

For example, $\frac{{5^6}}{{5^3}} = 5^{6 - 3} = 5^3$.

3. Power Rule

The power rule states that when raising an exponential expression to another exponent, you can multiply the exponents:

$(a^m)^n = a^{m \times n}$

For instance, $(3^2)^4 = 3^{2 \times 4} = 3^8$.

4. Zero Exponent Rule

The zero exponent rule states that any non-zero base raised to the power of zero is equal to 1:

$a^0 = 1$

For example, $7^0 = 1$.

5. Negative Exponent Rule

The negative exponent rule stipulates that any non-zero base raised to a negative exponent can be written as the reciprocal of the base raised to the positive exponent:

$a^{-n} = \frac{1}{{a^n}}$

For example, $2^{-3} = \frac{1}{{2^3}} = \frac{1}{8}$.

6. Product of Powers Rule

The product of powers rule states that when multiplying multiple exponential expressions with the same base, you can add all the exponents:

$a^m \times b^m = (a \times b)^m$

For instance, $2^3 \times 3^3 = (2 \times 3)^3 = 6^3$.

7. Power of a Power Rule

The power of a power rule dictates that when raising an exponential expression to another exponent, you can multiply the exponents:

$(a^m)^n = a^{m \times n}$

For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.

Conclusion

Make sure you practice, practice, practice these rules! They seem simple at first, but it can get very difficult if you don't master them!