Why do fractional exponents result in the square root operator?

In summary, raising a number to the 1/2 power is equivalent to finding its square root. This is because raising a number to the 1/2 power is the inverse operation of squaring. This relationship holds true for all positive real numbers, and can be generalized as x^(1/2) = √x. Understanding this relationship can help develop an intuitive understanding of algebraic operations.
  • #1
hackedagainanda
52
11
Homework Statement
What does 9^(1/2) = to?
Relevant Equations
9^`1 = 9
9^0 = 1
(x^a)(x^b)= x^a+b)
So I got the answer through a little addition i.e 9^(1/2) multiplied by 9^(1/2) = 9^1 or 9

3 x 3 = 9 so 3 is the answer to what is 9^(1/2)

I've tested this out with a few other numbers and have made this generalization, x^(1/2) = √x

It seems to make the equations orderly and consistent but is it by definition or is there a reason why raising it to a fractional exponent gives you this answer. I'm trying to not just memorize this rule, so I can get a better intuitive grasp on how the algebra works.
 
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  • #2
hackedagainanda said:
Homework Statement:: What does 9^(1/2) = to?
Relevant Equations:: 9^`1 = 9
9^0 = 1
(x^a)(x^b)= x^a+b)

So I got the answer through a little addition i.e 9^(1/2) multiplied by 9^(1/2) = 9^1 or 9

3 x 3 = 9 so 3 is the answer to what is 9^(1/2)

I've tested this out with a few other numbers and have made this generalization, x^(1/2) = √x

It seems to make the equations orderly and consistent but is it by definition or is there a reason why raising it to a fractional exponent gives you this answer. I'm trying to not just memorize this rule, so I can get a better intuitive grasp on how the algebra works.
##9^{1/2} = \sqrt 9 = 3##
 
  • #3
This is because (at least for positive real numbers)

$$(x^2)^{1/2} = x^{2/2} = x$$

Because in general ##(x^a)^b = x^{ab}##.
So raising to the 1/2 power performs the inverse operation of squaring, and hence it is the square root operator as you discovered.
 
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Related to Why do fractional exponents result in the square root operator?

What is a fractional exponent?

A fractional exponent is a way to represent a power or root that is not a whole number. It is written in the form of a fraction, with the numerator being the power and the denominator being the root.

How do you simplify a fractional exponent?

To simplify a fractional exponent, you can rewrite it as a radical expression. For example, 2^(1/2) can be rewritten as the square root of 2. You can also use the rules of exponents to simplify, such as multiplying the exponents when raising a power to a power.

Can a fractional exponent be negative?

Yes, a fractional exponent can be negative. This indicates that the base should be raised to the power and then take the reciprocal of the result. For example, 2^(-1/3) is equivalent to 1/(cube root of 2).

What is the difference between a fractional exponent and a radical?

A fractional exponent and a radical express the same mathematical operation, but in different forms. A fractional exponent is written as a fraction, while a radical is written using a root symbol. They can be converted to each other using exponent and root rules.

How do you solve equations with fractional exponents?

To solve equations with fractional exponents, you can use the rules of exponents to simplify the expressions and then solve the resulting equation. You may also need to use algebraic techniques, such as isolating the variable and using inverse operations, to solve for the variable.

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