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Simplify (7³ × 7² × 7⁶)/(7³ × 7⁵ × 7² × 7²).
Simplify seven to the third power times seven to the second power times seven to the sixth power all divided by seven to the third power times seven to the fifth power times seven to the second power times seven to the second power.
We can either evaluate each seven to the designated power and then multiply each of these numbers together and then divide or, when multiplying like bases, we can add their exponents. So on the numerator, we will take seven to the three plus two plus six power. And now for the denominator, it will be seven to the three plus five plus two plus two power.
So now we have a seven to the 11th power divided by seven to the 12th power. Now the reason why it’s nice to work with exponents, in these cases, is now that we’re dividing instead of taking seven to the 11th power and getting a very large number and seven to the 12th power and getting a very large number and then dividing, we can actually use a trick or property for our exponents when dividing.
We subtract them. And 11 minus 12 is negative one. Now when we have negative exponents, we can take them and move them either on the denominator or the numerator depending on where they’re already at. So since it’s seven to the negative first powers on the numerator, we need to move it down to the bottom, to the denominator. And that exponent will turn positive.
So we have one over seven to the positive first power, which is equal to one- seventh. Now let’s try solving it the other way, so we’re going to evaluate each of these numbers. Seven cubed is 343. Seven squared is 49. And seven to the sixth power is 117649. Seven to the fifth power is 16807. And then each of these seven squareds are 49.
Now before multiplying all of these numbers together on the numerator and the denominator, some of them can cancel. The 343s can cancel. And one pair of the 49s can cancel. And we end up with 117649 divided by 823543, which is indeed equal to one-seventh. So either route would work.