# Why X^(1/2) = root X and X^(-1/2) = 1/root x?

#### Indranil

##### Junior Member
Why X^(1/2) = root X and X^(-1/2) = 1/root x?

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#### mmm4444bot

##### Super Moderator
Staff member
Why X^1/2 = root X and X^-1/2 = 1/root x?

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

----------------

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:

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#### HallsofIvy

##### Elite Member
As mmm4444bot said, that is the way they are defined.

And the reason negative powers are defined that way is to keep some very nice properties of powers true even for non-positive integer powers.

For example, it is very basic to say that "$$\displaystyle a^2$$" is defined as "a multiplied by itself": $$\displaystyle a^2= (a)(a)$$. And "$$\displaystyle a^3$$" just adds another a to that $$\displaystyle a^3= (a)(a)(a)$$. So $$\displaystyle (a^2)(a^3)= ((a)(a))((a)(a)(a))= a^5$$. It is easy to see that, for m and n any positive integers, $$\displaystyle (a^m)(a^n)= a^{m+ n}$$. That's a very nice, easy to use, property!

But what does "$$\displaystyle a^0$$" mean? We can't multiply a number by itself 0 times! Well, we would like to have $$\displaystyle (a^m)(a^n)= a^{m+n}$$ true even when m or n are 0. That means we would like to have $$\displaystyle (a^m)(a^0)= a^{m+ 0}$$. But m+ 0= m so we must have $$\displaystyle (a^m)(a^0)= a^m$$. As long as a is not 0, we can divide on both sides by $$\displaystyle a^m$$ and have $$\displaystyle a^0= 1$$. That is, if we want $$\displaystyle (a^m)(a^n)= a^{m+n}$$ true even when n= 0, then we must define $$\displaystyle a^0$$ to be 1.

And what about negative powers? Well, n+ (-n)= 0 so we would want $$\displaystyle (a^n)(a^{-n})= a^{n+ (-n)}= a^0= 1$$. That is, in order to have that nice property still true, even for negative powers, we must define $$\displaystyle a^{-n}$$ to be the reciprocal of $$\displaystyle a^n[/b].$$

#### Indranil

##### Junior Member

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

----------------

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:
'a^(1/n) = 16^(1/4) = the fourth root of 16 = 2' Could please you simply ' the fourth root of 16'?

#### Subhotosh Khan

##### Super Moderator
Staff member
Why X^1/2 = root X and X^-1/2 = 1/root x?
Have you learned the rule of multiplication with exponents - i.e. -

am * an = a(m+n)

#### Subhotosh Khan

##### Super Moderator
Staff member
'a^(1/n) = 16^(1/4) = the fourth root of 16 = 2' Could please you simply ' the fourth root of 16'?
It seems that you are not familiar with "exponents".

Are you trying to "self-teach" algebra?

That will be a very difficult and time-consuming thing to do!!

#### HallsofIvy

##### Elite Member
'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.

#### Indranil

##### Junior Member
Have you learned the rule of multiplication with exponents - i.e. -

am * an = a(m+n)
Yes I have learned

#### Indranil

##### Junior Member
$$\displaystyle 2^2= 2(2)= 4$$. $$\displaystyle 2^3= 2(2^2)= 2(4)= 8$$. $$\displaystyle 2^4= 2(2^3)= 2(8)= 16$$. 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.
If 2^3 = 8, 8^1/3 = 2?

#### Jomo

##### Elite Member
Why X^1/2 = root X and X^-1/2 = 1/root x?
Just like + means to add and - means to subtract, a negative sign in a power means to take reciprocal (if you like)

#### mmm4444bot

##### Super Moderator
Staff member
If 2^3 = 8, 8^1/3 = 2?
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:

#### mmm4444bot

##### Super Moderator
Staff member
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?

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#### MarkFL

##### Super Moderator
Staff member
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?
I've been wondering about these issues as well. #### Indranil

##### Junior Member
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?
I am preparing for the medical entrance exam for 2019. It needs basic mathematics, Trigonometric, calculus, graph, etc for physics and chemistry.
So kind of you to promise to help me. Really, I appreciate all of your kind efforts which are helping me to learn math in an easy way that's why I will be grateful to all of you forever. Thanks a lot.

#### mmm4444bot

##### Super Moderator
Staff member
… 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?
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.

The thumbs-up icon indicates that you approve, you're happy, you like, etc.

The thumbs-down icon indicates that you don't approve, you're not happy, you don't like, etc.

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 …
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.)

Otherwise, if you're trying to learn it all for the first time, then I think it's best to follow a structured order because math instruction builds upon itself. That is, we often need to understand previous knowledge before we advance to new material. For example, the rules and properties in beginning algebra allow us to simplify expressions and solve equations. Later, we use those skills to learn about trigonometry, precalculus, and calculus. So, it makes sense to study math in order. (After all, we would not want a surgical intern slicing open our body first and then asking for help finding basic anatomy.) #### Indranil

##### Junior Member
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?
1. 2 root 4 = 4^1/2 = 2
2. ∛8 = 8¹/³ = 3
3. ∜16 = 16¹/⁴ = 2

#### mmm4444bot

##### Super Moderator
Staff member
Are these same thing below?
2 root 4 = 4^1/2 = 2

∛8 = 8¹/³ = 3
∜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.

2 root 4 -- Some people will not recognize that 2 is the index; instead, they will read it as 2 times root(4). For square roots, the standard form is sqrt(4). ['sqrt' is the name of the square-root function sqrt(x).]

Always type grouping symbols around the radicand.

4^1/2 -- We must enclose rational exponents within grouping symbols: 4^(1/2). When the grouping symbols are missing, the expression can be read as (4^1) divided by 2.

Where did you find the indexed radicals signs ∛ and ∜ ?

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#### JeffM

##### Elite Member
Dr. Peterson has already hinted at why exponents work the way they do: it is a process of generalization. But the laws of exponents are quite few and easy to memorize.

$$\displaystyle \text {Given: } a > 0,\ m,\ n \in \mathbb Z, \text { and } m \ge 0 \le 0:$$

$$\displaystyle \text {I: } a^0 \equiv 1;$$

$$\displaystyle \text {II: } m > 0 \implies a^m \equiv a * a^{(m-1)};$$

$$\displaystyle \text {III: } a^m * a^n \equiv a^{(m+n)};$$

$$\displaystyle \text {IV: } (a^m)^n \equiv a^{(mn)};$$

$$\displaystyle \text {V: } a^{(-n)} \equiv \dfrac{1}{a^n} \equiv \left ( \dfrac{1}{a} \right )^n;$$

$$\displaystyle \text {VI: } a^{(m - n)} = \dfrac{a^m}{a^n}, \text { and}$$

$$\displaystyle \text {VII: } a^{(m/n)} \equiv \sqrt[n]{a^m}.$$

Until you get to calculus, that is all you need. In fact you do not need to memorize even all of those because you can derive some of them from the others, but it is convenient to memorize them.

#### Indranil

##### Junior Member
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.

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
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?

#### MarkFL

##### Super Moderator
Staff member
The link is below where I found the indexed radicals signs ∛ and
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?
This site offers $$\displaystyle \LaTeX$$ powered by MathJax. For example the following markup:

Code:
$$\sqrt[n]{a^m}=a^{\frac{m}{n}}$$
produces:

$$\displaystyle \sqrt[n]{a^m}=a^{\frac{m}{n}}$$