Video Transcript
Find the geometric mean of nine 𝑥
to the 36th power and 36𝑦 to the 40th power.
Remember, if 𝑎 and 𝑏 are two
numbers which have the same sign, then their geometric mean is the square root of 𝑎
times 𝑏. Now, if the numbers have different
signs, then the product of 𝑎 and 𝑏 is negative, and so the geometric mean is
undefined. So, let’s take a closer look at the
two algebraic expressions we have. The first is the product of two
positive numbers. We know this because 𝑥 to the 36th
power has an even power, so substituting any real number into this expression will
give a positive output. And our next is also the product of
two positive numbers. 𝑦 to the 40th power has an even
power, and so it’s going to be nonnegative.
So, we can simply substitute the
expressions nine 𝑥 to the 36th power and 36 𝑦 to the 40th power in for 𝑎 and 𝑏,
respectively. And so, the geometric mean is the
square root of nine 𝑥 to the 36th power times 36𝑦 to the 40th power. And we could at this stage multiply
nine and 36 and then the algebraic expression. But the product of nine and 36 is
quite a big number. So, instead, we can use the laws of
radicals to separate each expression. And when we do, we see that it’s
equal to the square root of nine times the square root of 𝑥 to the 36th power times
the square root of 36 times the square root of 𝑦 to the 40th power. Then we know that the square root
of nine is three, and the square root of 36 is six. Three times six is 18, so the
coefficient of our final expression is going to be 18.
But how do we evaluate the square
root of 𝑥 to the 36th power? Well, of course, the square root of
some real number — let’s call that 𝑐 — can be written as 𝑐 to the power of
one-half. Then, we’re finding 𝑥 to the 36th
power to the power of a half. And to simplify this, we multiply
the exponents; 36 times one-half is 18. So, the square root of 𝑥 to the
36th power is 𝑥 to the 18th power. We repeat this with the square root
of 𝑦 to the 40th power. It’s 𝑦 to the 40th power to the
power of a half. And then, 40 times one-half is
20. So, the square root of 𝑦 to the
40th power is 𝑦 to the 20th power. And so, we have our geometric mean;
it’s 18𝑥 to the 18th power times 𝑦 to the 20th power.
