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
The properties of radicals allow expressions with radicals to be rewritten.
Simplify the expression using the properties of radicals. 53⋅15
In order to simplify an nth root, it is necessary that the radicand can be expressed as a power. If the index of the radical and the power of the radicand are equal, nan, the radical expression can be simplified as follows. nan={∣a∣ if n is odd∣a∣ if n is even
The absolute value of a number is always non-negative, so when n is even, the result will always be non-negative.Next, rewrite the radicals using rational exponents. The example can be rewritten as follows. ∣x∣⋅6x4x3⋅x⋅3x⇔∣x∣⋅x61x43⋅x⋅x31
When the expression has been simplified completely, it can be rewritten using radicals. ∣x∣x1211=∣x∣12x11 This is the simplified expression.
Simplify the expression using the properties of exponents. 434⋅16312310