Question

Simplify the expression. The simplified expression should have no negative exponents. $$ x^{5} \cdot \frac{1}{x^{8}} $$

Short answer

The simplified expression with no negative exponents for \(x^{5} * 1/x^{8}\) is \(1 / x^{3}\).

Step-by-step solution

Recognize the division of exponents

The given expression is a division of two numbers with exponents. When you divide two numbers with the same base, the rule of exponents states that you subtract the exponent of the denominator from that of the numerator. In this case, both numbers have the base of \(x\), so you can apply the rule: \(x^{n} / x^{m} = x^{(n-m)}\). Thus the given expression \(x^{5} * 1/x^{8}\) simplifies to \(x^{(5-8)}\).

Simplify the exponent

Now simplifying the exponent inside the brackets: \(x^{(5-8)} = x^{-3}\). This results in a negative exponent.

Convert to positive exponent

As per the rules of exponents, a negative exponent can be converted to a positive exponent by taking its reciprocal. Thus, \(x^{-3}\) can be written as \(1 / x^{3}\).