Question Video: Finding Geometric Mean of Two Algebraic Expressions Mathematics • 10th Grade

Find the geometric mean of 9𝑥³⁶ and 36𝑦⁴⁰.

02:25

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.

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