Problem 29 - College Algebra Essentials

Question

Simplify each exponential expression. $$x^{-5} \cdot x^{10}$$

Step-by-Step Solution

Answer

The simplified expression of \(x^{-5} \cdot x^{10}\) is \(x^5\).

Step 1: Understanding the Properties of Exponents

An important property of exponents states that for any non-zero number, \(a\) and any integers \(m\) and \(n\), \(a^m \cdot a^n = a^{m+n}\). Essentially, when multiplying two expressions with the same base, we keep the base and add the exponents.

Step 2: Applying the Property

In this example, both expressions have the same base, \(x\). So, apply the property to simplify the expression: \(x^{-5} \cdot x^{10} = x^{-5 + 10}\).

Step 3: Simplify the Exponent

Add the exponents together: \(x^{-5 + 10} = x^5\).