[SPLIT, MOVED] Radical notations

[marciedrewryan]

I'm having a hard time grasping this stuff. I've been out of school for over 15 years, and all of this is a blur.... This is what I'm stuck on... 8 to the power of -2/3

 

[stapel]

In future, kindly please post new questions as new threads, rather than resurrecting years-old threads. Thank you.

I'm having a hard time grasping this stuff. I've been out of school for over 15 years, and all of this is a blur.... This is what I'm stuck on... 8 to the power of -2/3

Are you needing lessons to help with exponents in general, with negative exponents, with fractional exponents, or with some other aspect of the posted expression? We'll be glad to provide you with lesson links so that you can refresh yourself on what you'd studied so long ago, but some direction would likely be helpful.

 

[JeffM]

I'm having a hard time grasping this stuff. I've been out of school for over 15 years, and all of this is a blur.... This is what I'm stuck on... 8 to the power of -2/3

The laws of exponents start with some definitions.

\(\displaystyle [1] \ a^0 = 1.\)

\(\displaystyle [2] \ a^1 = a.\)

\(\displaystyle [3] \ a^b * a^c = a^{(b + c)}.\)

\(\displaystyle [4] \ \left(a^b\right)^c = a^{(b * c)}.\)

\(\displaystyle [5] \ a^b = a^c \implies b = c.\)

From these definitions, certain other laws can be derived.

\(\displaystyle \ b\ is\ a\ positive\ whole\ number\ > 1 \implies a^b = a^1 * a^{(b - 1)} \implies\)

\(\displaystyle [6]\ a^b = product\ of\ a\ times\ itself\ b\ times.\)

The law above is usually given as the definition of exponents when you first learn about them. It has been found highly useful to define exponents more generally as shown above. It is hard to understand rational exponents if you keep thinking about them as positive whole numbers. It is better to start with statements 1 through 5 as defining exponents. Then the old definition for exponents still is true, but it is seen to be a special case that is a consequence of the expanded definition.

\(\displaystyle \ a * a^x = 1 \implies a^1 * a^x = a^0 \implies a^{1 + x} = a^0 \implies 1 + x = 0 \implies x = - 1 \implies a^1 * a^{-1} = a^0 \implies a * a^{-1} = 1 \implies\)

\(\displaystyle [7]\ a^{-1} = \dfrac{1}{a}.\)

\(\displaystyle [9]\ a^{-b} = \left(a^b\right)^{-1} = \dfrac{1}{a^b}.\)

In other words, \(\displaystyle 8^{-(2/3)} = \dfrac{1}{8^{(2/3)}}.\)

\(\displaystyle b\ is\ a\ positive\ whole\ number > 1\ and\ a^x = \sqrt{a} \implies \left(a^x\right)^b = \left(\sqrt{a}\right)^b \implies a^{(bx)} = a = a^1 \implies bx = 1 \implies x = \dfrac{1}{b} \implies\)

\(\displaystyle [10]\ a^{(1/b)} = \sqrt{a}.\)

\(\displaystyle [11]\ a^{c/b} = \left(a^c\right)^{(1/b)} = \sqrt{a^c}.\)

Those eleven laws of exponents will let you solve (almost ?) any problem involving exponents.

 

[mmm4444bot]

This is what I'm stuck on ... 8 to the power of -2/3

We use parentheses and the caret symbol ^ to denote exponentiation.

8^(-2/3)

You forgot to post the instructions that came with this expression. Are they asking you to rewrite the given expression using radical notation?

Please check the summary page of our posting guidelines, for information about how to best ask for help on these boards. Thank you!

 

[lookagain]

The laws of exponents start with some definitions.

\(\displaystyle [1] \ a^0 = 1. \ \ \)a not equal to 0.

\(\displaystyle [5] \ a^b = a^c \implies b = c. \ \ \) a could equal -1 or 1, and then b not equal to c.


\(\displaystyle \ a * a^x = 1 \implies a^1 * a^x = a^0 \implies a^{1 + x} = a^0 \implies 1 + x = 0 \implies x = - 1 \implies a^1 * a^{-1} = a^0 \implies a * a^{-1} = 1 \implies \ \ \) a not equal to 0.

\(\displaystyle [7]\ a^{-1} = \dfrac{1}{a}. \ \ \) a not equal to 0

\(\displaystyle [9]\ a^{-b} = \left(a^b\right)^{-1} = \dfrac{1}{a^b}. \) a not equal to 0
\(\displaystyle [11]\ a^{c/b} = \left(a^c\right)^{(1/b)} = \sqrt{a^c}. \ \ \) not both a = 0 and c = 0.

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