Why X^1/2 = root X and X^-1/2 = 1/root x?
'a^(1/n) = 16^(1/4) = the fourth root of 16 = 2' Could please you simply ' the fourth root of 16'?
It's because of definitions.
There's a property of exponents that defines the meaning of a negative sign in front of an exponent. The negative sign denotes the reciprocal of the power.
a^(-n) = 1/a^n
EG:
In the power a^n, let the base a = 2 and the exponent n = 3, so we have the power 2^3 = 8.
If this exponent becomes negative, then we have the reciprocal of the power.
2^(-3) = 1/(2^3) = 1/8
There's another property in algebra that defines the meaning when an exponent is a fraction.
a^(m/n) = the nth root of a^m
In your example, m = 1, so the property simplifies to
a^(1/n) = the nth root of a
EGs:
Let a = 9 and n = 2
a^(1/n) = 9^(1/2) = the square root of 9 = 3
a^(-1/n) = 9^(-1/2) = the reciprocal of the square root of 9 = 1/3
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Let a = 16 and n = 4
a^(1/n) = 16^(1/4) = the fourth root of 16 = 2
a^(-1/n) = 16^(-1/4) = the reciprocal of the fourth root of 16 = 1/2
----------------
Let x = 49 and n = 2
x^(1/2) = 49^(1/2) = 7
x^(-1/2) = 49^(-1/2) = 1/7
PS: It's good form in algebra to not interchange upper- and lower-case symbols. Use X, or use x, but don't use both to represent the same thing. :cool:
Have you learned the rule of multiplication with exponents - i.e. -Why X^1/2 = root X and X^-1/2 = 1/root x?
It seems that you are not familiar with "exponents".'a^(1/n) = 16^(1/4) = the fourth root of 16 = 2' Could please you simply ' the fourth root of 16'?
\(\displaystyle 2^2= 2(2)= 4\). \(\displaystyle 2^3= 2(2^2)= 2(4)= 8\). \(\displaystyle 2^4= 2(2^3)= 2(8)= 16\). The "roots" go the opposite way: The square root of 4 is 2, the cube root of 8 is 2, and the fourth root of 16 is 2.'a^(1/n) = 16^(1/4) = the fourth root of 16 = 2' Could please you simply ' the fourth root of 16'?
Yes I have learnedHave you learned the rule of multiplication with exponents - i.e. -
am * an = a(m+n)
If 2^3 = 8, 8^1/3 = 2?\(\displaystyle 2^2= 2(2)= 4\). \(\displaystyle 2^3= 2(2^2)= 2(4)= 8\). \(\displaystyle 2^4= 2(2^3)= 2(8)= 16\). The "roots" go the opposite way: The square root of 4 is 2, the cube root of 8 is 2, and the fourth root of 16 is 2.
Just like + means to add and - means to subtract, a negative sign in a power means to take reciprocal (if you like)Why X^1/2 = root X and X^-1/2 = 1/root x?
Yes.If 2^3 = 8, 8^1/3 = 2?
So far, you have posted four thumbs-down signs, in this thread. Are you unhappy with some of our responses? If so, can you explain why, so that we can better understand what you're looking for?
I have looked over your posts since February. You have posted basic questions about a lot of topics (beginning algebra, intermediate algebra, trigonometry, precalculus, calculus). Learning all of these topics takes a few years. You seem to be jumping back and forth in only a few months, without a clear purpose.
What specifically are you trying to accomplish here? Are you studying for some type of placement test? Are you currently taking some math course(s)? Are you playing around for fun but not seriously?
We can provide better suggestions, if we understand what you're tying to do. :cool:
EDIT: I just noticed that you put a thumbs-down sign in your OP's subject line. Why did you do that?
Please don't get me wrong. Actually, I don't know what 'the thumb down' and 'the thumb up' symbols mean. I only use those symbols to attract your attention, nothing else. Then could you suggest me which symbols should I use?So far, you have posted four thumbs-down signs, in this thread. Are you unhappy with some of our responses? If so, can you explain why, so that we can better understand what you're looking for?
I have looked over your posts since February. You have posted basic questions about a lot of topics (beginning algebra, intermediate algebra, trigonometry, precalculus, calculus). Learning all of these topics takes a few years. You seem to be jumping back and forth in only a few months, without a clear purpose.
What specifically are you trying to accomplish here? Are you studying for some type of placement test? Are you currently taking some math course(s)? Are you playing around for fun but not seriously?
We can provide better suggestions, if we understand what you're tying to do. :cool:
EDIT: I just noticed that you put a thumbs-down sign in your OP's subject line. Why did you do that?
You don't need to add symbols, to attract our attention. This forum uses bold-face type, to indicate unread posts for us, and it has a New Posts button. Most of the regular volunteers here see all new, unread posts.… Actually, I don't know what 'the thumb down' and 'the thumb up' symbols mean. I only use those symbols to attract your attention, nothing else. Then could you suggest me which symbols should I use?
Thanks for this information. Have you studied these topics before? That is, are you refreshing your memory on some points, but you recognize most of what you see at your study site? I ask because, if you're refreshing your memory, then perhaps it's not bad that you're jumping back and forth within the topics. (You said that you're learning math in an easy way here; that's good. You're the ultimate judge as to whether you're ready for exams.)I am preparing for the medical entrance exam for 2019. It needs basic mathematics, Trigonometric, calculus, graph, etc for physics and chemistry … I appreciate all of your kind efforts which are helping me to learn math in an easy way …
Are these same thing below?Yes.
2 is the third root of 8 because we need three factors of 2 to make 8.
8 = 2 ∙ 2 ∙ 2
Likewise 2 is the fourth root of 16 because we need four factors of 2 to make 16.
16 = 2 ∙ 2 ∙ 2 ∙ 2
How many factors of 3 do we need to make 243? We need five of them. This means that 3 is the fifth root of 243.
243 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3
What if somebody asked, "What's the fifth root of 243?" and we did not know that it's 3. In other words, how could we write the fifth root of 243, if we did not know it was 3?
We could write it like this: 243^(1/5).
Likewise, we could express the fourth root of 16 as 16^(1/4) and the third root of 8 as 8^(1/3). :cool:
Are these same thing below?
2 root 4 = 4^1/2 = 2
∛8 = 8¹/³ = 3
That 3 is a typo, yes?∜16 = 16¹/⁴ = 2
That 3 is a typo, yes?
If I understand your question (i.e., you're asking whether the expressions are equal as shown), then I say YES.
Here are some comments about notation.
2 root 4 -- Some people may not recognize that the 2 is meant to be an index; instead, they might read this as 2 times root 4. For a square root, the standard is sqrt(4). Always type grouping symbols around the radicand.
4^1/2 -- We always need to enclose rational exponents within grouping symbols: 4^(1/2). If the grouping symbols are missing, the expression could be read as 4^1 divided by 2.
Where did you find the indexed radicals signs ∛ and ∜?
The link is below where I found the indexed radicals signs ∛ and ∜
http://usefulwebtool.com/en/math-keyboard.php
Could you provide me with a tool or site where I can write scientific notations properly as the keyboard does not have any such symbols?
[tex]\sqrt[n]{a^m}=a^{\frac{m}{n}}[/tex]