How to Simplify Expressions with Exponents
In this video, we will put all of the exponent properties we have learned together. Don't let lots of numbers and letters confuse you! Simplifying exponents can be easy and fun!
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Review of the Properties of Exponents
Let’s review the exponent properties:
Product of Powers : ( x ^ a )( x ^ b ) = x ^ a+b
Power to a Power : ( x ^ a )^ b = x ^ a b
Quotient of Powers : ( x ^ a ) / ( x ^ b ) = x ^ a-b
Power of a Product : ( xy )^ a = x ^ a y ^ a
Zero Property : x ^0 = 1
If you’re having problems memorizing these properties, I suggest using flash cards. Flash cards are a fantastic and easy way to memorize topics, especially math properties.
Split up these expressions using multiplication after putting like terms on top of one another
Practice Problem #1
So, let’s get started!
Simplify is the same as reducing to lowest terms when we talk about fractions. Simplifying these terms using positive exponents makes it even easier for us to read.
Our first expression has x ^3 y ^8 / y ^3 x ^7. The first step I like to do is put the like terms on top of each other. On the top, I have x ^3 y ^8. In the denominator, I want the x s over each other and the y s over each other, so I write x ^7 y ^3.
My next step is to split these up using multiplication. This step is important when you first begin because you can see exactly what we are doing. Splitting the multiplication gives us x ^3 / x ^7 times y ^8 / y ^3.
Next step - look at each part individually. Since we have x ^3 divided by x ^7, we subtract their exponents. This gives us x ^3-7. Since we have y ^8 divided by y ^3, we subtract their exponents. This gives us y ^8-3. This will give us x ^3-7, which is -4 and y ^8-3, which is 5.
Solution to the first practice problem
Remember, we’re simplifying using positive exponents, so we need to change x ^-4. We know from our exponent properties that x ^-4 is 1 / x ^4 times y ^5. Well, 5 is positive, so we don’t need to change it.
My last step is to multiply. Our final, simplified answer is y ^5 / x ^4. This is our simplified answer with positive exponents.
Practice Problem #2
Let me show you another one. This time we have 5 x ^2 y ^9 / 15 y ^9 x ^4. Let’s rewrite this with like terms over each other: 5/15 times x ^2 / x ^4 times y ^9/ y ^9
We start at the beginning. 5/15 reduces to 1/3. Next, x ^2 divided by x ^4 is x ^(2-4). y ^9 divided by y ^9 is y ^(9-9). Let’s keep simplifying. We have 1/3 times x ^(2-4), which is -2, times y ^(9-9), which is y ^0. This gives us 1/3 times 1/ x ^2 times 1. Multiplying straight across, our final answer is 1/3 x ^2.
Steps in solving practice problem #2
This is our answer simplified using positive exponents.
Practice Problem #3
There are a lot of letters and numbers here, but don’t let them trick you. If we keep separating the terms and following the properties, we’ll be fine.
Our first step is to simplify (2 p )^3. We distribute the exponent to everything in the parenthesis. This will give us (8 p )^3 q ^4 in the bottom or denominator, but our top or numerator will stay the same.
Next, we separate them into multiplication: 16/8 times p / p ^3 times q ^2 / q ^4 times r ^9.
Here’s the fun part, simplify. 16/8 is 2/1 times p ^(1-3) times q ^(2-4) times r ^9.
We’re almost done: 2 times p ^(1-3) is -2, times q ^(2-4), which is q ^(-2) times r ^9.
We are asked to simplify using positive exponents: p ^(-2) is the same as 1/ p ^2; q ^(-2) is the same 1/ q ^2.
Solution for the third practice problem
Finally, our last step - multiplying the fractions straight across. Our final answer is r ^9 / p ^2 q ^2. This is in simplified form using positive exponents.
Remember, it will take time and practice to be good at simplifying fractions.
After this lesson you’ll be able to simplify expressions with exponents.
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