Is three to the 74th power over
three to the 75th power greater than, less than, or equal to three?
Let’s begin this question by
thinking about what three to the 74th power and three to the 75th power will
actually involve. We can recall that, for example,
three to the fourth power means that we write three down four times and multiply it
together. So three to the 74th power means
that there would be 74 lots of three multiplied together. Three to the 75th power would be 75
lots of three multiplied together. Actually, calculating the value of
either of these would take a very long time. And in this question, we also need
to divide. But let’s have a look at what
happens if we cancel values from the numerator and denominator.
We could cancel off the first
threes and the second two threes and so on until we have canceled off 74 threes on
the numerator and the denominator, leaving us with just three on the
denominator. We know that this will be
equivalent to one-third. So three to the 74th power over
three to the 75th power is equal to one-third. There is another way we could’ve
calculated this, though. The quotient law of exponents tells
us that 𝑥 to the power of 𝑎 over 𝑥 to the power of 𝑏 is equal to 𝑥 to the power
of 𝑎 minus 𝑏.
Here, the value of 𝑥 would be
three and our two exponents 𝑎 and 𝑏 would be 74 and 75. So our answer here would be three
to the power of 74 minus 75. And that’s equivalent to three to
the power of negative one. Three to the power of negative one
is the reciprocal of three, and that’s the same as one-third. Our question asks if this is
greater than, less than, or equal to three. One-third is less than three. And so that’s our answer.