The nth root of a number a expresses another number that, when multiplied by itself n times, will result in a. Aside from the radical symbol, the notation is made up of the radicand a and the index n.
The resulting number is commonly called a radical. For example, the radical expression 416 is the fourth root
of 16. 416 simplifies to 2 because 2 multiplied by itself 4 times equals 16.
416=424=2
The general expression na represents a number which equals a when multiplied by itself n times.
n timesna⋅na⋅…⋅na=aor(na)n=a
The nth root can also be expressed using a rational exponent.
na1=an1
When a number is raised to the power of a fraction, that fraction is the number's rational exponent. Such an expression is equivalent to a root.
anm=nam
Rewrite the given expressions in the opposite form. 5xandx32
To begin, notice that the expression 5x is written in radical form. Thus, it needs to be rewritten with a rational exponent. Recall that nam=anm. Notice that in 5x, x does not have an exponent. That means it is raised to the power of 1. This gives 5x=x51.
This expression has a rational exponent. Thus, we must rewrite it with a radical. Recall that anm=nam. Since the denominator of the exponent is 3, we can write the cube root of x2. x32=3x2
Simplify the following radical expression by rewriting it so that it has a rational exponent. 11533