Video: AQA GCSE Mathematics Foundation Tier Pack 1 β€’ Paper 1 β€’ Question 17

Which of the following expressions does not simplify to π‘˜β΄? Circle your answer. [A] (π‘˜ Γ— π‘˜β΅)/π‘˜Β² [B] π‘˜βΈ Γ· π‘˜β΄ [C] π‘˜ Γ— π‘˜ Γ— π‘˜ Γ— π‘˜ [D] π‘˜βΈ/π‘˜Β²

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Video Transcript

Which of the following expressions does not simplify to π‘˜ to the power of four? Circle your answer. The options are π‘˜ multiplied by π‘˜ to the power of five over π‘˜ squared, π‘˜ to the power of eight divided by π‘˜ to the power of four, π‘˜ multiplied by π‘˜ multiplied by π‘˜ multiplied by π‘˜, and π‘˜ to the power of eight divided by π‘˜ squared.

First, let’s begin by recalling what is meant by π‘˜ to the power of four. π‘˜ to the power of four means π‘˜ multiplied by π‘˜ multiplied by π‘˜ multiplied by π‘˜. That’s π‘˜ multiplied together four times. We can see straightaway that this is actually one of the options we’ve been given. So we know that this expression does simplify to π‘˜ to the power of four.

In order to decide whether the other options simplify to π‘˜ to the power of four, we need to recall some of our index laws. The first law tells us what to do if we’re multiplying together two powers of the same base. In this case, the base is π‘˜. If we have π‘˜ to the power of π‘Ž multiplied by π‘˜ to the power of 𝑏, then this is equal to π‘˜ to the power of π‘Ž plus 𝑏. We add the powers together.

We can demonstrate why this rule works by using an example. Let’s consider π‘˜ cubed multiplied by π‘˜ squared π‘˜ cubed means π‘˜ multiplied by π‘˜ multiplied by π‘˜, π‘˜ multiplied together three times. And then, we’re multiplying by π‘˜ squared. That’s π‘˜ multiplied by π‘˜.

Let’s count how many times we’re multiplying π‘˜ together overall: one, two, three, four, five. So we see that π‘˜ cubed multiplied by π‘˜ squared is equal to π‘˜ to the power of five. And we know that five is equal to three plus two. That’s the sum of the powers of π‘˜ that we were originally multiplying. So this illustrates the first rule.

The second rule tells us what to do if we are dividing powers of the same base. In this case, we subtract the powers. π‘˜ to the power of π‘Ž divided by π‘˜ to the power of 𝑏 is equal to π‘˜ to the power of π‘Ž minus 𝑏. We can again see why this works using example.

Let’s consider π‘˜ cubed divided by π‘˜ squared. We have π‘˜ multiplied by π‘˜ multiplied by π‘˜ β€” that’s π‘˜ cubed β€” over π‘˜ multiplied by π‘˜; that’s π‘˜ squared. But we can cancel factors of π‘˜ in the numerator and denominator of this fraction. One π‘˜ here will cancel with a π‘˜ in the denominator and then a second π‘˜ in the numerator will cancel with a second π‘˜ in the denominator. We’re left with just π‘˜. But π‘˜ means π‘˜ to the power of one. And notice that one is equal to three minus two. So we’ve also illustrated the second rule.

We’ll now use these two rules to simplify each of the remaining three expressions in turn. Let’s begin with π‘˜ multiplied by π‘˜ to the power of five over π‘˜ squared. We can think of π‘˜ as π‘˜ to the power of one. We can then apply the first rule to simplify the numerator. π‘˜ to the power of one multiplied by π‘˜ to the power of five is π‘˜ to the power of one plus five. That’s π‘˜ to the power of six.

We can then apply our second rule. We have π‘˜ to the power of six divided by π‘˜ squared. So that’s π‘˜ to the power of six minus two. Six minus two is equal to four. So we can see that this first expression does indeed simplify to π‘˜ to the power of four.

Now, let’s consider the second expression: π‘˜ to the power of eight divided by π‘˜ to the power of four. By our second rule, that’s π‘˜ to the power of eight minus four. Eight minus four is equal to four. So the second expression also simplifies to π‘˜ to the power of four.

We’ve already shown that the third expression is indeed equal to π‘˜ to the power of four. So we’re only left with one expression at this point: π‘˜ to the power of eight over π‘˜ squared. Let’s show then that it does not simplify to π‘˜ to the power of four.

By applying our second rule, we need to subtract the powers. π‘˜ to the power of eight over or divided by π‘˜ squared is equal to π‘˜ to the power of eight minus two. Eight minus two is equal to six. So this fourth expression doesn’t simplify to π‘˜ to the power of four; it simplifies to π‘˜ to the power of six. We, therefore, circle this final expression as our answer.

This final answer does represent a common misconception which is to think that when we are dividing, we also need to divide the powers. π‘˜ to the power of eight divided by π‘˜ squared is sometimes mistakenly thought of as π‘˜ to the power of eight divided by two which would indeed be equal to π‘˜ to the power of four. However, as we’ve seen, this is incorrect. If we are dividing, then we need to subtract, not divide the powers. Watch out for this common mistake.

We’ve shown that the expression that does not simplify to π‘˜ to the power of four is π‘˜ to the power of eight over π‘˜ squared.

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