Question Video: Evaluating Expressions Involving Laws of Exponents Mathematics • 8th Grade

Is 3⁷⁴/3⁷⁵ greater than, less than, or equal to 3?

02:06

Video Transcript

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.

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