Calculate four to the power of a
half squared multiplied by nine to the power of a half squared.
There are two ways we can answer
this question. The first is to consider what we
actually mean by a power of one-half or an index of one-half. In fact, a fractional index gives a
root. And the denominator of the fraction
tells us which root we’re looking for. So the power of a half is a square
root. And the power of a third or the
index a third gives us a cube root.
In this case, both four and nine
have a power of one-half. So we’re square-rooting each of
these numbers inside the bracket. We could at this point work out the
value of the square root since both four and nine are square numbers. However, each bracket is being
squared. Squaring is the inverse or the
opposite of square-rooting. So the square root of four squared
is just four, and the square root of nine squared is nine. So our sum becomes four multiplied
by nine, which is simply 36.
An alternative method we could’ve
considered would have been to use the laws of indices. When we have a bracket with an
index, we can multiply the powers. So 𝑥 to the power of 𝑎 to the
power of 𝑏 is the same as 𝑥 to the power of 𝑎 multiplied by 𝑏. This means four to the power of a
half all squared is four to the power of a half multiplied by two. And a half multiplied by two is
Similarly, nine to the power of a
half all squared is nine to the power of a half multiplied by two, which is nine to
the power of one. And in fact, anything to the power
of one is just itself, so we have four and nine. Once again, we multiply these as
per the original sum and we get four multiplied by nine to be 36. Either method is valid here.