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Course: Algebra 1>Unit 12

Lesson 7: Exponential vs. linear models

Linear vs. exponential growth: from data (example 2)

Sal constructs a function that models cooling water. To do that, he decides whether the function is linear or exponential.

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• Why is the common ratio raised to the 1/2 power the same as the square root of itself?
• Hi!

I think this can best be shown with a proof. It's actually quite simple.

Let the common ratio be r, and let's let r^(1/2)=y.

So:
r^(1/2) = y
By squaring both sides, we get:
r = y^2
Now we take the square root of both sides:
sqrt(r) = y
And since we know that y = r^(1/2), we can therefore conclude:
r^(1/2) = sqrt(r)
• @ Sal uses a fraction in the exponent to solve the problem. His answer does not match the solutions provided and how he does it makes no sense at this point. He completely skipped over how the fraction on the exponent is supposed to work and he's basically using topics that haven't been covered to solve a problem, without explaining why it works. How am I supposed to solve a problem with a tool I don't know how to use?
• Sal uses one of the exponential properties. Heres the link (this unit is after learning abt exponential properties):

The question wants the answer in terms of t by itself. When you evaluate the question, you get this formula as shown in the video:

80(.8)^(t/2)

t isn't by itself, it's divded by 2. Now we use one of the properties.

Refer to this: x^(ab) = ((x^a)^b)
We can take out t/2 by using this formula. In multiplication form, t/2 is simply just 1/2 * t.

Using this rule, set a = 1/2 || b = t || x = .8
((.8^(1/2))^t)

.8^(1/2) is simply saying sqrt(.8) (also another exponential property), and that equals roughly .9, and A is your answer.

hopefully that helps !
• How would you define "model"? Thank you !
• Hey Ramon M, there are two different definitions from the dictionary for model (that are nouns), I hope these help!

1: A three-dimensional representation of a person or thing or of a proposed structure, typically on a smaller scale than the original.

2: A system or thing used as an example to follow or imitate.
• hi, why take the square root is the same thing as making its exponent 1/2? like he did at:
• A square root is is A number times that number, right? So if you have a square root of a number, but want to still express that number as an exponent, you put that exponent to half. here is an examle, because i am not good at explaining.

√25 =5
25^1/2 =5
try it out! i hope this helps
• Why is the common ratio raised to the 1/2 power the same as the square root of itself?
• Why is the common ratio raised to the 1/2 power the same as the square root of itself?
(1 vote)
• Why is the common ratio raised to the 1/2 power the same as the square root of itself?
• Would there perhaps be a more intuitive way of thinking about why a common ration raise to the 1/2 power is the same as the square root of itself?
Kelvin L.’s answer is great but it would otherwise help remembering it better.
• At about , Sal writes 80^1/2^t. I don't understand how he gets there from 80^t/2. How are they the same thing?
• Sal took out the t from the exponent. Since t/2 could be seen as 1t/2, you can just make the exponent 1/2^t
(1 vote)
• Sorry I'm confused. Is the temperature decreasing by 80% or 20%. Intuitively, I feel it is reducing by 20%. Am I missing something?