Chapter 8.1, Problem 51HP

To determine

Tocalculate:

Two pairs of whole numbers with geometric mean.

Answer to Problem 51HP

  $36$ and $4$ , $81$ and $4$ are two pairs of whole numbers with a geometric mean that is also a whole number.

Explanation of Solution

Given information:

Geometric mean that is also whole number.

Calculation:

The geometric mean of two positive numbers $a$ and $b$ is $x=\sqrt{ab}$ by the definition of the geometric mean.

Now fid the geometric mean of $36$ and $4$ .

So by the definition of the geometric mean,

  $x=\sqrt{ab}$

Now substitute,

  $\begin{array}{l}x=\sqrt{36\left(4\right)}\\ =\sqrt{144}\\ =12\end{array}$

Hence, the geometric mean of $36$ and $4$ is $12$ .

Now, find the geometric mean of $81$ and $4$ .

So, by the definition of the geometric mean,

  $x=\sqrt{ab}$

Now substitute,

  $\begin{array}{l}x=\sqrt{81\left(4\right)}\\ =\sqrt{324}\\ =18\end{array}$

Hence, the geometric mean of $81$ and $4$ is $18$ .

Therefore, $36$ and $4$ , $81$ and $4$ are two pairs of whole numbers with a geometric mean that is also a whole number.

In order for $\sqrt{ab}$ to be a whole number, $ab$ must be a perfect square. So, the geometric mean of two whole numbers is a whole number if and only if the product of the numbers is a perfect square.