Question

Using the ratio of perfect squares method, what is the square root of 96 rounded to the nearest hundredth?

Answer 1

When the square root is a whole number, it is called a perfect square. For instance, 25 is a perfect square whose square root is 5.

We have a list of perfect squares that we can generate by squaring the natural numbers, e.g., $1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25$. Now, there are two perfect squares in particular that we want to look at: $9^2=81$ and $10^2=100$.

Why? Well, 96 is between 81 and 100, therefore, it's square root is between 9 and 10. In fact, since 96 is a lot closer to 100, its square root is closer to 10.

If we want to approximate it, we first guess. Knowing its square root is closer to 10, we can guess that it is 9.6. Let's check the square of 9.6.
$9.6^2=9.6 \times 9.6 = 92.16\\92.16\ \textless \ 96$
The square of 9.6 is less than 96, so we try a bigger number. Let's try 9.7.
$9.7^2=9.7 \times 9.7 = 94.09\\94.09\ \textless \ 96$
We're closer, but not quite there yet. Let's try 9.8.

$9.8^2=9.8 \times 9.8 = 96.04\\96.04\ \textgreater \ 96$
We are very close now, but our answer seems a little bit too big. We know we can't go down to 9.7 again, so we go out to a second decimal place. Let's try 9.78.
$9.78^2=9.78 \times 9.78 = 95.65 \\ 95.65 \ \textless \ 96$
We want something bigger, so let's try 9.79.
$9.79^2=9.79 \times 9.79 = 95.84 \\ 95.84 \ \textless \ 96$

Now it looks like we need to go up again, and while we could now move out to the thousandth place, your question asks us to round to the hundredths place. So it looks like 9.8 was accurate after all, but rounded to the hundredths place, it's 9.80.

We can confirm on a calculator that indeed the square root of 96 is about 9.80.