Part a) Calculate the square root of seven and one-ninth as an improper fraction.
An improper fraction is one with the numerator of the fraction is greater than the denominator. So this suggests that before we can calculate the square root of seven and one-ninth, our first step will be to convert seven and one-ninth from a mixed number into an improper fraction.
Each whole number is equal to nine-ninths. So if there are nine-ninths in one, then in seven there will be seven times as many ninths. That’s sixty-three ninths. Adding one to this for the extra ninth gives sixty-four ninths. So we’ve converted seven and one-ninth from a mixed number to an improper fraction.
Now, we need to find its square root. We can apply a key property of square roots or surds here, which is that the square root of a fraction is actually equal to the square root of the numerator over the square root of the denominator. So the square root of 64 over nine can be written as the square root of 64 over the square root of nine.
64 and nine are both square numbers. So their square roots are integers. The square root of 64 is eight and the square root of nine is three. This is an improper fraction because the numerator of eight is greater than the denominator of three. So we’ve calculated the square root of seven and one-ninth as eight over three.
Part b) says, “Write the fifth root of 81 as a power of three.”
To write something as a power of three means to write it in the form three to some power which we’ll need to work out. First, we’ll think of an alternative way of writing a fifth root. A root like this is actually a fractional root. The 𝑛th root of 𝑎 is equal to 𝑎 to the power of one over 𝑛. So a fifth root is the power of one-fifth.
We can, therefore, write the fifth root of 81 as 81 to the power of one-fifth. Now, remember we want to write this as three to some power. So next, we’re going to try and convert 81 into some power of three.
You may not know what 81 is as a power of three off by heart. But you probably do know that 81 is equal to nine squared. And you probably know that nine is equal to three squared. So we have three squared squared to the power of one-fifth. That’s nine squared giving 81 to the power of one-fifth.
To simplify this further, we need to recall another one of our rules of powers, which is that if we have a number or letter to some power and then raised to another power, we multiply the powers together. So if we have 𝑎 to the power of 𝑚 and then to the power of 𝑛, we have 𝑎 to the power of 𝑚𝑛. Now, it’s 𝑚 multiplied by 𝑛. So here, we have three to the power of two and then two again. So that’s three to the power of two times two.
We’re then raising this to the power of a fifth. So we’re also multiplying the power by one-fifth, giving three to the power of two times two times one-fifth. Two times two is four. And by writing four as four over one, if it helps, then we can see that four multiplied by one-fifth is four-fifths because we multiply the numerators together to give four and the denominators together to give five. So our answer simplifies to three to the power of four-fifths.
You may have been able to get to this answer more quickly if you know that 81 is actually equal to three to the power of four. Three multiplied by three gives nine, multiplying by three again gives 27, and multiplying by three again gives 81.
You could then apply that second law of powers that we discussed, so multiplying the two powers together, to give three to the power of four multiplied by one-fifth. As before, the power simplifies to four-fifths.
So we’ve written the fifth root of 81 as a power of three. It’s equal to three to the power of four-fifths.