# Fractional Exponents (How is it done?)

• Bored
In summary, when raising a power to another power, you can split up the exponent and multiply the base by each part of the exponent separately. For example, 2^5/2 can be split up into 2^2 * 2^1/2, which is equivalent to 4 * √2. This is because when multiplying powers, you add the exponents. This method can also be used with fractional exponents, where you can split up the exponent into an integer exponent and a fraction with a coefficient less than one, in order to keep the root in the equation.
Bored
How does 2^5/2 become 2^2 multiplied by 2^1/2?
(The '^' means 'to the power of' so 2 to the power of 5/2. I am not sure how to write this as an exponent as this is my first post.)

2^5/2 = 2^2 × 2^1/2

So 2^2 = 4 and 2^1/2 means Square Root so there is a radical sign, so it becomes √2.

I tried to reverse engineer the solution but I'm still not sure how 2^5/2 makes it. I know you first Square root it so √2 then you put to the power of 5 so √2^5. This goes into decimals so I am confused on what to do next. I looked at the solution and I can see that both of them make 2^5/2 I just don't know how it was factorized into those specific numbers, because 4 x √2 = 4√2 and somehow that is √2^5? I know 2^5 also makes 32.

So I think I might of done something wrong or I don't know the correct method, can someone please attempt to help me and thank you for your time!

Do you agree that x^(a+b)=(x^a) (x^b) for natural x,a,b?

edit, forgot a +

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In my case would that be 2^5/2 (5 is a and 2 is b) = (2^5) (5^2)? I'm quite sure I'm doing something wrong so I think I can't agree on that even though I think its correct.

So what is a and b? If a and b are 5 and 2 they make (2^5) (5^2) making 800. I am kind of lost... Am I getting it wrong?

Another go at it I see that 2^5 = 32 and 5^2 = 25 - So do I put in the fraction making it 2^5/1 and 5^1/2?

Oh now I see it and the '+', Thank you for changing it

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Do you agree that
$2^4 = 2^2 * 2^2$
$3^{10}=3^4 * 3^6$

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Yes, Because when you are multiplying powers you add them together, but since I started fractional exponents I am very confused - probably because of the fractions.

So if you added 2^2 and 2^1/2 that would make 2^3/2 if i am correct or 2.5?

Okay so there's 2/1 + 1/2 = 2 5/2
So adding fractions you double it to make it 4/2 then add it to make it 5/2 making it 2^5/2.
So we done this backwards but how did 2^5/2 split up to make those 2?

Oh wait (Edit), So you could also spilt it up to something else? But it has to be a Fractional Exponents because so it can be combined? Not sure why it has to be a fractional exponent but I know you split the expression into the product of an integer exponent and the factor with a fractional coefficient less than one. I think the reason why this is done is so you can keep the root in the equation, otherwise it no longer has a fractional exponent.

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$2+\frac{1}{2}=\frac{4}{2}+\frac{1}{2}=\frac{5}{2}$

2^5/2 is problematic, it could mean ##\frac{2^5}{2}##. You can write it as 2^(5/2) or better [noparse]25/2[/noparse], which gets parsed as 25/2. And you can use LaTeX here.

You can split up the 5/2 like this:
$$\frac{5}{2}=\frac{4+1}{2}=\frac{4}{2}+\frac{1}{2}=2+\frac{1}{2}$$
And then use the rule for exponents written above.

## 1. What is a fractional exponent?

A fractional exponent is a mathematical expression that represents the root of a number. It is written in the form of a power with a numerator and denominator, where the numerator is the power and the denominator is the root. For example, 21/2 represents the square root of 2.

## 2. How do you simplify a fractional exponent?

To simplify a fractional exponent, you can use the rule that states: am/n = (a1/n)m. This means that you can take the root of the base and raise it to the power of the numerator. For example, 82/3 can be simplified to (81/3)2 = 22 = 4.

## 3. What is the difference between a fractional exponent and a radical?

A fractional exponent and a radical are two ways of representing the same mathematical concept. A fractional exponent is a power with a numerator and denominator, while a radical is the root symbol with a number inside. For example, 21/2 and √2 both represent the square root of 2.

## 4. Can a fractional exponent be negative?

Yes, a fractional exponent can be negative. A negative exponent represents the reciprocal of the number raised to the positive exponent. For example, 2-1/2 is equal to 1/21/2, which is the same as √1/2 or 1/√2.

## 5. How do you solve equations with fractional exponents?

To solve equations with fractional exponents, you can use the rule that states: am/n = (a1/n)m. This means that you can take the root of the base and raise it to the power of the numerator on both sides of the equation. For example, to solve for x in the equation 22/3 = x, you can take the cube root of both sides to get x = 22/3 = (21/3)2 = 22/3 * 2 = 24/3.

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