Problem 127 - College Algebra

Question

Why is \(\left(-3 x^{2}\right)\left(2 x^{-5}\right)\) not simplified? What must be done to simplify the expression?

Step-by-Step Solution

Answer

So, the simplified form of the given expression \(\left(-3 x^{2}\right)\left(2 x^{-5}\right)\) is \(-6x^{-3}\).

Step 1: Recognize the Constants and Exponents

In this expression, you have two parts: \(-3x^2\) and \(2x^{-5}\). In both parts you can see a constant and an \(x\) term with an exponent. The first part has the constant \(-3\) and the \(x\) term \(x^2\). The second part has the constant \(2\) and the \(x\) term \(x^{-5}\).

Step 2: Multiply the Constants

You first multiply the constants -3 and 2. The multiplication of -3 and 2 gives -6. So, the simplified form of the constants will be -6.

Step 3: Multiply x terms

According to the exponent rule when you multiply terms with the same base, you add the exponents. Given \(x^2\) and \(x^{-5}\), the base is \(x\) and the exponents that need to be added are 2 and -5. Adding 2 and -5 gives -3. So, the simplified form of the \(x\) terms is \(x^{-3}\).

Step 4: Combine Simplified Constants and x terms

Combine the simplified constants and \(x\) terms from steps 2 and 3 to get your final simplified expression. Multiplying the constants -6 with the \(x\) term \(x^{-3}\) gives \(-6x^{-3}\).