Rational Exponents

Rational = Fraction

What does it mean to have a fraction in the exponent?

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We know the are of a square is length \times width, in this case length and width are the same, so length^{2}=s. To find out the length we need to use the inverse operation, the squareroot. So length=\sqrt{s}

This makes sense because \sqrt{s} \sqrt{s} = \sqrt{s^2} = s

(or \sqrt{s} \sqrt{s} = \sqrt{s}^2=s they’re both the same.)

Using our properties of exponents we can see that if we write

    \[s^{\frac{1}{2}} \cdot s^{\frac{1}{2}}=s\]

since we add the exponents. From here we learn that \sqrt{s}=s^{\frac{1}{2}}

Having a fraction in the exponent is called a RATIONAL EXPONENT. A rational exponent is just another way to write a root.

 

\sqrt[m]{b^{n}}=b^{\frac{n}{m}}