Negative 40𝑥 to the ninth power divided by what equals negative five 𝑥 to the seventh power?
Before we begin to answer this question, let’s look at how we would approach a similar calculation with simpler values. Let’s say we have the calculation 15 divided by what equals five. In this case, our missing number and our value five are both factors of 15. In order to calculate our missing factor, we would simply take our 15 and divide it by the factor that we know, five, which would of course give us the answer three, since three times five is 15.
Returning to the original question then, we can see that there are two factors of negative 40𝑥 to the ninth power. One of them is negative five 𝑥 to the seventh power, and the other one we need to work out. So to find our missing value, we can rearrange the calculation into the more solvable one, negative 40𝑥 to the ninth power divided by negative five 𝑥 to the seventh power.
We can approach this calculation in two different methods, either by using straight division or by considering it as a fraction. So let’s begin by considering it as a division. Here we have our dividend, negative 40𝑥 to the ninth power, within our division lines and our divisor, negative five 𝑥 to the seventh power, on the outside. When we consider the division of our coefficients, negative 40 divided by negative five, this will give us the answer eight, since the division of two negative values will always give us a positive value.
Next, to divide our powers of 𝑥, we need to remember the rules for division with exponents. If we have a value 𝑥 raised to the power of 𝑎 divided by 𝑥 raised to the power of 𝑏, the answer will be 𝑥 raised to the power of 𝑎 minus 𝑏. This means that 𝑥 to the ninth power divided by 𝑥 to the seventh power will be 𝑥 squared. And so our answer to the division calculation negative 40𝑥 to the ninth power divided by negative five 𝑥 to the seventh power is eight 𝑥 squared.
And as a second alternative method, we can consider our calculation as a fraction. So in this case, our numerator will be negative 40𝑥 to the ninth power and our denominator will be negative five 𝑥 to the seventh power. As this is a fraction, we can work through our calculation by simplifying the numerator and denominator or thinking of it in terms of cancelling down.
In the same way as before, we can simplify our coefficients negative 40 over negative five as eight, giving us eight 𝑥 to the ninth power over 𝑥 to the seventh power. And we can simplify our powers of 𝑥, 𝑥 to the ninth power over 𝑥 to the seventh power, as 𝑥 squared, giving us an answer of eight 𝑥 squared. So using either the division method or the fraction method will give us the missing answer in the question as eight 𝑥 squared.