2.10 Combine laws to simplify expressions
Introduction
We've previously looked at all of the power laws. Now we'll look at common occurences of combining multiple laws in order to solve more complex problems.
Ideas
Combine multiplication laws and the power of a power law
Sometimes we will need to use the multiplication laws and the power of a power law together.
Examples
Example 1
Simplify the expression (2^{3})^{5} \times 2^{4}.
When we need to use multiple product power properties, consider the expression from left to right and apply the appropriate properties as you need them.
Power of a power rule with negative exponents or bases
We already learned to apply the power of a power rule to positive exponents. This rule states: (x^{a})^{b}=x^{a\times b}
Now we are going to explore what happens when we also include negative values in these kinds of questions.
If you think back to learning about multiplying and dividing by negative numbers, you'll remember that:
Multiplying a positive and a negative number will give you a negative answer.
Multiplying two negative numbers together gives you a positive answer.
Let's think for a minute about what happens if we multiply a negative value by itself more than twice.
If we have (-2)^{3}, this means (-2) \times (-2) \times (-2).
If we simplify this, (-2) \times (-2) = 4 and 4\times (-2)= -8.
What about if we multiplied it by itself again to get the answer to (-2)^{4}?
We know that (-2)^{3}=-8 and -8\times (-2) =16.
So, as a general rule:
If you put a negative number to an even power, you will get a positive answer.
If you put a negative number to an odd power, you will get a negative answer.
Examples
Example 2
Simplify (2^{2})^{-3}.
If you put a negative number to an even power, you will get a positive answer.
If you put a negative number to an odd power, you will get a negative answer.
The power of a power rule with negative exponents works the same way as with positive exponents, we just have the added step of applying the negative power property at some point in our working.
General combination of exponent properties
It's important to remember the order of operations when solving questions that require multiple exponent properties being used.
We may encounter an expression that requires the combination of exponent properties of the form (a^{m}\times b^{n})^{p}, and we can use a combination of the multiplication and power of a power properties to see that (a^{m}\times b^{n})^{p}= a^{m\times p}\times b^{n\times p}
Examples
Example 3
Write the following with a positive exponent: 3^9 \div 3^5 \times 3^4.
Example 4
Simplify (2^{2}\times3^{-1})^{-2}.
Summary of exponent properties:
The multiplication property: a^{m} \times a^{n}=a^{m+n}
The division property: a^{m} \div a^{n}=\dfrac{a^{m}}{a^{n}}=a^{m-n},\,a^{n} \neq 0
The power of a power property: (a^{m})^{n}=a^{m\times n}
The zero power property: a^{0}=1, \, a\neq 0
When an expression requires us to use multiple properties, we consider the expression from left to right using order of operations and the properties as necessary.