Simplify 144𝑥 to the 20th plus 168𝑥 to the 17th plus 84𝑥 to the 12th over 12𝑥 to the 12th plus three 𝑥 to the eighth power plus 19𝑥 to the fifth power plus 18.
Our first step here will be to look at our fraction and see if there’s anything we can simplify. Almost immediately, we can notice that 144, 168, and 84 are all divisible by 12. In addition to that, they all have an 𝑥 term. We’ll deal with their exponents in just a minute.
First let’s divide each of these whole numbers by 12: 144 divided by 12 is 12; 168 divided by 12 equals 14; 84 divided by 12 equals seven. When we divide exponents, we take the exponent and the numerator and we subtract the exponent from the denominator.
In order to divide 𝑥 to the 20th by 𝑥 to the 12th, we’ll say 𝑥 to the 20 minus 12. To divide 𝑥 to the 17 by 𝑥 to the 12th, we’ll say 𝑥 to the 17 minus 12. And finally, for the third term, we’ll say 𝑥 to the 12 minus 12 power.
Let’s go ahead and solve these subtraction problems: 20 minus 12 equals eight, 17 minus 12 equals five, and 12 minus 12 equals zero, but remember that anything to the zero power is one, which means we’ll just have seven times one here. And when we simplify that, it’s just seven.
But we have three more terms that have just been hanging out up there, so we wanna bring those down. In this step, we should check and see, are there any like terms? Are there any terms we can combine? 12𝑥 to the eighth and three 𝑥 to the eighth are like terms; we can combine them.
To combine these like terms, we simply add their coefficients; 12 plus three equals 15. And we’re finished with those. From there, we see that 14𝑥 to the fifth and 19𝑥 to the fifth are like terms and can be combined. By combining their coefficients, we get 33𝑥 to the fifth, and we’re finished with those.
Finally, we have two whole numbers that can also be combined. Seven plus 18 equals 25. Our simplified expression would look like this: 15𝑥 to the eighth plus 33𝑥 to the fifth plus 25.