Fractional Exponents
Square roots are most often written using a radical sign, like this: `sqrt4`. But there is another way to represent the taking of a rootAny number `x` multiplied by itself a specific number of times to produce another number, such that in `x^n = y`, `x` is the `n`th root of `y` - for example, because `2^3 = 8`, `2` is the `3`rd (or cube) root of `8`. . We can use fractional exponents instead of a radicalThe math symbol `sqrt`, used to denote the process of taking a root of a quantity .
Can’t imagine raising a number to a fractional power? They may be hard to get used to, but fractional exponents can actually help simplify some problems. Let’s see how these fractional exponents we call rational radicals work.
Radicals and exponents are inverse operations. It may be surprising, then, to learn that a radical can be expressed as an exponential number. The table below shows some examples of common square roots written as radicals, fractional exponents, and integers. Notice that the denominator of the fractional exponent is the number `2`.
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Radical |
Exponent |
Integer |
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`sqrt16` |
`16^(1/2)` |
`4` |
|
`sqrt25` |
`25^(1/2)` |
`5` |
|
`sqrt100` |
`100^(1/2)` |
`10` |
Let’s look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in these examples, the denominator of the fractional exponent is the number `3`.
|
Radical |
Exponent |
Integer |
|
`root(3)(8)` |
`8^(1/3)` |
`2` |
|
`root(3)(125)` |
`125^(1/3)` |
`5` |
|
`root(3)(729)` |
`729^(1/3)` |
`9` |
These examples help us model a relationship between radicals and fractional exponents: namely, that the `n`th root of a number can be written as either `root(n)(x)` or `x^(1/n)`.
|
Radical |
Exponent |
|
`sqrtx` |
`x^(1/2)` |
|
`root(3)(x)` |
`x^(1/3)` |
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`root(4)(x)` |
`x^(1/4)` |
|
… |
… |
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`root(n)(x)` |
`x^(1/n)` |
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The `5`th root of the number `243` can be written as either:
A) `root(5)(243)` or `243^(1/5)`
B) `root(1/5)(243)` or `243^5`
C) `root(243)(5)` or `5^(1/243)`
D) `root(5)(243)` or `243^5`
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All of the numerators for the fractional exponents in the examples above were `1`. We can use exponents other than unit fractions to express roots, as shown below. Notice any patterns within this table?
|
Radical |
Exponent |
|
`sqrt9` |
`9^(1/2)` |
|
`root(3)(9^2)` |
`9^(2/3)` |
|
`root(4)(9^3)` |
`9^(3/4)` |
|
`root(5)(9^2)` |
`9^(2/5)` |
|
… |
… |
|
`root(n)(9^x)` |
`9^(x/n)` |
To write a radical as a fractional exponent, the power to which the base is raised becomes the numerator and the root becomes the denominator.
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Writing Fractional Exponents
Any radical in the form `root(n)(a^x)` can be written as a fractional exponent in the form `a^(x/n)`.
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This makes sense for our unit fraction exponents as well. For example, the radical `sqrt81` can also be written as `sqrt(81^1)`, since any number remains the same value if it is raised to the first power. We can now see where the numerator of `1` comes from in the equivalent form of `81^(1/2)`.
Fractional exponents are not used very much, outside of advanced formulas in upper-level math and science. But occasionally they are useful in simplifying algebraic expressions.
Let’s explore some radical expressionsA quantity that contains a term with a radical, as in `2sqrt3` or `root(3)(8a^3bc^6)`. now and see how. Here’s a radical expression that needs simplifying, `root(3)(a^6)`.
One method of simplifying this expression is to factor and pull out groups of `a^3`, as shown below:
`root(3)(a^6)`
`root(3)(a^3*a^3)`
`root(3)(a^3)*root(3)(a^3)`
`a*a`
`a^2`
We can also simplify this expression by thinking about the radical as a fractional exponent, and using the principle that any radical in the form `root(n)(a^x)` can be written as a fractional exponent in the form `a^(x/n)`:
`root(3)(a^6)`
`a^(6/3)`
`a^2`
Note that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.
Both simplification methods gave us the same result, `a^2`, Depending on the context of the problem, it may be easier to use one method or the other, but for now, we’ll note that we were able to simplify this expression more quickly using fractional exponents than we were using the “pull-out” method.
Let’s try a more complicated expression, `(10b^2c^2)/(croot(3)(8b^4))`. Wow! This expression has two variables, a fraction, and a radical. That’s a little intimidating. We’ll take it step-by-step and see if using fractional exponents can help us simplify it.
We’ll start by simplifying the denominator because this is where the radical sign is located.
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Example |
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Problem |
Simplify: `(10b^2c^2)/(croot(3)(8b^4))` |
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`(10b^2c^2)/(c*root(3)(8)*root(3)(b^4))` |
Separate the terms in the denominator. |
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`(10b^2c^2)/(c*2*root(3)(b^4)` |
Take the cube root of `8`, which is `2`. |
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`(10b^2c^2)/(c*2*b^(4/3))` |
Rewrite the radical as an exponent. |
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`10/2*c^2/c*b^2/b^(4/3)` |
Rewrite the fraction as a series of factors in order to cancel terms (see next step). |
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`5*c*b^2/b^(4/3)` |
Simplify the constant and `c` terms. |
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`5cb^2b^(-4/3)` |
Use the rule of negative exponents, `n^-x=1/n^x`, to rewrite `1/(b^(4/3))` as `b^(-4/3)`. |
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`5cb^(2/3)` |
Combine the `b` terms by adding the exponents. |
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`5croot(3)(b^2)` |
Change the exponent back to a radical. By convention, an expression is not usually considered simplified if it has a fractional exponent or a radical in the denominator. |
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Answer |
`5croot(3)(b^2)`
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Well, that took a while, but we did it. We applied what we know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.
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Simplify:`(4a^2)/(2sqrta)`
A) `2a`
B) `2a^3`
C) `2asqrta`
D) `1/(4root(3)(a)`
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A radical can be expressed as a value with a fractional exponent by following the convention `root(n)(a^x)=a^(x/n)`. Rewriting radicals as fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.