Trouble with radicals

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stormbytes

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Screen Shot 2020-07-25 at 2.40.58 PM.jpg

That's an excerpt from my textbook.

I understand the following:

1) 9-2 should be the same as 91/2

2) 91/2 following the formula this should equal 2-root(91) which should = 3

However when I try this on a calculator (Ti-83+) I get all kinds of crazy.

Where am I going wrong?
 
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stormbytes said:
View attachment 20646

That's an excerpt from my textbook.

I understand the following:

1) 9-2 should be the same as 91/2

2) 91/2 following the formula this should equal 2-root(91) which should = 3

However when I try this on a calculator (Ti-83+) I get all kinds of crazy.

Where am I going wrong?
Click to expand...
You say:

1) 9-2 should be the same as 91/2

NO ..... certainly not - What statement/s in your textbook gave you that idea?

The following will be correct:

\(\displaystyle 9^{-2} \ = \ \frac{1}{9^2} \ = \ \frac{1}{81} \)

Now start all over again and stop driving your calculator crazy.
 
Subhotosh Khan said:
You say:

1) 9-2 should be the same as 91/2

NO ..... certainly not - What statement/s in your textbook gave you that idea?

The following will be correct:

\(\displaystyle 9^{-2} \ = \ \frac{1}{9^2} \ = \ \frac{1}{81} \)

Now start all over again and stop driving your calculator crazy.
Click to expand...

Yeah I just got that ugh.

a-b = 1/ab

I was thinking -2 = 1/2 so I did that and then following the formula.
 
Ugh why do textbooks have to be so damned obtuse??

I totally don't follow the connection here...

Screen Shot 2020-07-25 at 8.40.20 PM.jpg
 
stormbytes said:
Ugh why do textbooks have to be so damned obtuse??

I totally don't follow the connection here...

View attachment 20648
Click to expand...
Connection between what and what?

If you are asking about the connection between

\(\displaystyle a^{\frac{1}{n}} \ \text and \ \ \sqrt[n]{a} \ \ \)

it is just a different way writing the same thing (like a synonym or writing the same word in different scripts).

Those are equal by definition and there is no "why" to it.
 
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I’d like to understand how they get from one to the other.

not simply fraction to radical but the actual example provided.
 
stormbytes said:
I’d like to understand how they get from one to the other.

not simply fraction to radical but the actual example provided.
Click to expand...
\(\displaystyle \left[64\right]^{\frac{1}{3}} \ = \ \sqrt[3]{64}\)

There is no "how" or "why" to it - it is just by definition. It is two different ways of writing the samething.
 
Subhotosh Khan said:
\(\displaystyle \left[64\right]^{\frac{1}{3}} \ = \ \sqrt[3]{64}\)

There is no "how" or "why" to it - it is just by definition. It is two different ways of writing the samething.
Click to expand...
I was referring to the example showing (a^1/n)^n = a^(1/n)n etc..

how are they getting that extra n?
What’s being shown here?
 
stormbytes said:
I was referring to the example showing (a^1/n)^n = a^(1/n)n etc..
how are they getting that extra n? What’s being shown here?
Click to expand...
This is being shown: \(\left(\sqrt[3]{x^4}\right)^6=\left(x^{4/3}\right)^6=x^8\) That is showing how exponent rules work.
 
stormbytes said:
I was referring to the example showing (a^1/n)^n = a^(1/n)n etc..

how are they getting that extra n?
What’s being shown here?
Click to expand...
What they're saying is that, whatever [MATH]x^{1/n}[/MATH] means, it's a number that, when raised to the nth power, yields [MATH](x^{1/n})^n = x^{1/n\cdot n} = x[/MATH]. And what is the number whose nth power is x? It's called the nth root of x!

So that's how we interpret [MATH]x^{1/n}[/MATH]. Technically, it hasn't been defined until we make this statement, but this reasoning shows that this is the only definition that will be compatible with known facts.
 
We want the laws of exponents to work for rational numbers.Therefore, they must translate into the laws of exponents for integers because integers are included within the rational numbers.

We have as a general law in the integers:

[MATH]a > 0 \text { and } (a^p)^q = a^{pq}.[/MATH]
So when we extend to the rationals we want to maintain that law

[MATH]x = a^{(1/p)} \implies \\ x^p = (a^{(1/p)})^p = a^{(p * 1/p)} = a^1 \implies \\ x^p = a \implies \\ x = \sqrt[p]{a}.[/MATH]
 
stormbytes said:
Yeah I just got that ugh.

a-b = 1/ab

I was thinking -2 = 1/2 so I did that and then following the formula.
Click to expand...
-2 is a negative number and 1/2 is a positive number. No negative number equals a positive number.
 
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