Powers Of Powers
Hi there. In this math education post, I talk about the algebra topic of powers to a power. This topic is normally a part of high school mathematics (grade 9).
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Topics
- Review Of Multiplying Monomials
- Power To A Power
- Practice Problems
- Solutions To Practice Problems
Review Of Multiplying Monomials
Before getting into learning about power to a power, it is best to review exponent laws when it comes to multiplying monomials.
Given two monomials such as 10x and 
, we can multiply them together. Then 10 and 2 multiply together to obtain 20 and x times x-squared gives x to the power of three.
Think of 
as the variable being multiplied by itself where there are three x's.
Power To A Power
Consider a monomial such as 5x but raised to the power of 2. This would like 
. What would this look like?
Let's go back to the definition of an exponent and apply exponent laws.
Having x-squared to the power of three gives x to the power of 6. The simpler way is to multiply the exponents of 2 and 3 to obtain the new exponent of 6.
The general formula given a base, exponent and a second exponent would be:
where p and q are (usually) integers.
A More Complex Example
So far we dealt with a single variable with a coefficient of 1. What if we have another variable and a number other than 1 in front? Let's take a look at 
.
The outside exponent of 5 is applied to the 2, 
and y in the bracket. Two the power of 5 gives 32 for the new coefficient, the new exponent for x is 15 from 3 times 5 and the exponent for y is 5 (1 times 5).
In general, the formula would be:
where a, b, c are different variables and q is an integer exponent.
Practice Problems
Try out these practice problems for better understanding and increasing algebra speed.
Solutions To Practice Problems
