# 5.4 Fractional exponents

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This module provides sample problems designed to develop concepts related to fractional exponents.

On the homework, we demonstrated the rule of negative exponents by building a table. Now, we’re going to demonstrate it another way—by using the rules of exponents.

• ## A

According to the rules of exponents , $\frac{{7}^{3}}{{7}^{5}}$ $7^{\mathrm{\left[ \right]}}$ .
• ## B

But if you write it out and cancel the excess 7s, then $\frac{{7}^{3}}{{7}^{5}}$ = ——.
• ## C

Therefore, since $\frac{{7}^{3}}{{7}^{5}}$ can only be one thing, we conclude that these two things must be equal: write that equation!

Now, we’re going to approach fractional exponents the same way. Based on our rules of exponents , $9^{\left(\frac{1}{2}\right)}^{2}$ =

So, what does that tell us about $9^{\left(\frac{1}{2}\right)}$ ? Well, it is some number that when you square it, you get _______ (*same answer you gave for number 2). So therefore, $9^{\left(\frac{1}{2}\right)}$ itself must be:

Using the same logic, what is $16^{\left(\frac{1}{2}\right)}$ ?

What is $25^{\left(\frac{1}{2}\right)}$ ?

What is $x^{\left(\frac{1}{2}\right)}$ ?

Construct a similar argument to show that $8^{\left(\frac{1}{2}\right)}=2$ .

What is $27^{\left(\frac{1}{3}\right)}$ ?

What is $-1^{\left(\frac{1}{3}\right)}$ ?

What is $x^{\left(\frac{1}{3}\right)}$ ?

What would you expect $x^{\left(\frac{1}{5}\right)}$ to be?

What is $25^{\left(\frac{-1}{2}\right)}$ ? (You have to combine the rules for negative and fractional exponents here!)

OK, we’ve done negative exponents, and fractional exponents—but always with a 1 in the numerator. What if the numerator is not 1?

Using the rules of exponents, $8^{\left(\frac{1}{3}\right)}^{2}=8^{\mathrm{\left[ \right]}}$ .

So that gives us a rule! We know what $8^{\left(\frac{1}{2}\right)}^{2}$ is, so now we know what 8⅔ is.

$8^{\left(\frac{2}{3}\right)}=$

Construct a similar argument to show what $16^{\left(\frac{3}{4}\right)}$ should be.

Check $16^{\left(\frac{3}{4}\right)}$ on your calculator. Did it come out the way you predicted?

Now let’s combine all our rules! For each of the following, say what it means and then say what actual number it is. (For instance, for $9^{\left(\frac{1}{2}\right)}$ you would say it means $\sqrt[]{9}$ so it is 3.)

$8^{\left(\frac{-1}{2}\right)}=$

$8^{\left(\frac{-2}{3}\right)}=$

For these problems, just say what it means. (For instance, $3^{\left(\frac{1}{2}\right)}$ means $\sqrt[]{3}$ , end of story.)

$10^{-4}$

$2^{\left(\frac{-3}{4}\right)}$

$x^{\left(\frac{a}{b}\right)}$

$x^{\left(\frac{\mathrm{-a}}{b}\right)}$

find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
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hmm well what is the answer
Abhi
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20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
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Abhi
🤔.
Abhi
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salma
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salma
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Sherica
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Sherica
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China
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Stotaw
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anybody can imagine what will be happen after 100 years from now in nano tech world
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silver nanoparticles could handle the job?
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not now but maybe in future only AgNP maybe any other nanomaterials
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I'm interested in Nanotube
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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