Part a) Work out 9.87 squared plus 31.7 multiplied by the cube root of 7.86. Write down all the figures on your calculator’s display.
Most scientific calculators will allow us to type this exact problem in and give us
the relevant answer. We do need to be a little bit aware of some of the quirks of these calculators
The chances are your calculator will look a little bit like this. There is a button that will allow us to square any number. So we can type 9.87 squared using this button.
And it is sensible to write down an intermediate step. In this case, we could write down the value of 9.87 squared to be 97.4169.
Scientific calculators will also apply BIDMAS. So we don’t need to worry about the fact that we have an addition and a
multiplication in the same problem.
And if you’re trying to find the cube root symbol, it’s usually near the square root
button. In this case, it’s the alternative function on the square root button. And to get to it, we press the shift button and then the square root button.
And if we were gonna write this as an intermediate step, we’d see that the cube root
of 7.86 is 1.9882 and so on. And this is why it’s often sensible to write the intermediate steps but then type the
original problem into your calculator exactly as shown.
If we were to use 1.9882 rounded to some extent, that might change the accuracy of
our final answer. In fact, if we follow the steps and type the problem in exactly as shown, we get
Notice that the question says to write all the numbers on the calculator’s
display. So we’re absolutely must not round our answer. Instead, we leave it as shown.
Part b) Use approximations to test whether your answer to part a is reasonable. Show all your working below.
When we make approximations, we estimate by rounding each part of the problem to one
significant figure. The first significant figure in any number is the first nonzero digit.
In the number 9.87, the first nonzero digit is the nine. The digit immediately to the right of this is sometimes called the deciding
digit. Remember if the deciding digit is less than five, we round the number down. And if it’s five or above, we round the number up.
This helps us decide in this case whether our answer is closer to the nine or the
10. And in fact, because eight is greater than five, 9.87 is closer to 10 than it is to
nine. And 9.87 rounds to 10.
The first nonzero digit in the second number 31.7 is the three. This means that the one immediately to its right is the deciding digit. Since one is less than five, this means that 31.7 is closer to 30 than it is to
40. And we round down. 31.7 correct to one significant figure is 30.
Our final number is 7.86. The first nonzero digit is the seven. And the deciding digit is once again an eight. We saw that the deciding digit being an eight means we round the number up. In this case, 7.86 is closer to eight than it is to seven. And 7.86 correct to one significant figure is eight.
Using these estimations and our calculation becomes 10 squared plus 30 minus the cube
root of eight. 10 squared is 10 multiplied by 10; it’s 100. And the cube root of eight is two since two multiplied by itself and multiplied by
itself again, two cubed, is eight.
BIDMAS or BODMAS tells us to multiply 30 by two first to get 60. And then, we add 160 to get 160.
Our answer to part a was just a little bit over 160. So our estimation of 160 is very close to the answer from part a. And it must be a reasonable solution.