Problem 110 - College Algebra Essentials

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\left(3 x^{-4} y z^{-7}\right)(3 x)^{-3}$$

Step-by-Step Solution

Answer

The simplified exponential expression form is \(3^2 y x^{-7} z^{-7}\).

Step 1: Rewrite the exponential multiplication

Rewrite \(\left(3 x^{-4} y z^{-7}\right)(3 x)^{-3}\) as \((3 x^{-4} y z^{-7})(3 x^{-3})\). Observe that it’s multiplying like bases – in this case, the base is \(x\). Follow the rule of multiplying bases by adding the exponents.

Step 2: Multiply the like bases

So now the expression becomes \((3 y z^{-7} x^{-4})(1 x^{-3})\). The coefficient of \(x^{-3}\) in the second parentheses is one since there is no number before it. Use the multiplication rule of exponents to add the powers of \(x\). This rule is \(x^m x^n = x^{m+n}\). Therefore, the powers of \(x\) can be simplified as \(x^{-4}x^{-3}=x^{-4-3}=x^{-7}\)

Step 3: FInal expression

Substituting everything into the expression, the final simplified form is \(3^2 y x^{-7} z^{-7}\). Notice that \(3^2\) is a result of \((3)\) times \((3)\). This is because both were part of the original functions, and now they are multiplied together.