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7.0 Radicals and Rational Exponents
An exponent is the number that tells us how many times to use the base as a factor.
When the exponent is 2, we call it squaring the number - we write the base twice and
multiply.
The whole numbers that we get as a result of squaring are called perfect squares.
The opposite of squaring, is taking the square root.
If you take the square root of a perfect square, you get an integer.
Except for zero, every perfect square has two square roots, a positive one and a negative
one. For example, (+4)(+4) = 16 and (– 4)( – 4) = 16. Consequently, 16 has two square
roots, both + 4 and – 4.
If we use a radical sign, or square root sign, to signify the square root, we mean only the
positive, or principal, square root.
81  9
If you want to use a radical sign and get both square roots, you have to use a plus or minus
sign to indicate that you want them both.
 121  11
The number under the radical sign is called the radicand.
The index is a small number in the crook of the radical to tell us what root we want. If there
is not an index showing, the root is understood to be the square root.
3
8 or
4
16
Since squaring a number and taking the square root of a number are opposite operations,
doing them both usually gets you back where you started.
 4
2
4
EXCEPTION: Watch out for negatives under the radical sign.
(4) 2  16  4
If you are taking the root of a fraction, just take the root of the top and put it over the root
of the bottom.
9
9 3


4
4 2
Irrational roots – If you take the square root of a number that is not a perfect square,
you’ll get a decimal. These are always non-terminating, non-repeating decimals – irrational
numbers. You can only approximate the value of an irrational number. If they ask for an
exact answer, they want the symbol 2 . If they ask for an approximate answer, they’ll have
to tell you how exact they want the approximation (nearest tenth, for example).
Square roots of negative numbers – NO CAN DO in the set of real numbers
16
NO SOLUTION
When we are looking for a square root, we are trying to find two identical twins that
multiply to give the radicand. (-3)2 = 9 so one of the square roots of 9 is – 3. To get a
negative answer by multiplying, the signs have to be different. If the signs are different, the
two numbers are not identical twins!