Problem 109 - College Algebra Essentials

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\left(2 x^{-3} y z^{-6}\right)(2 x)^{-5}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(y/16x^{8}z^{6}\).

Step 1 - Identify the base with its exponent

Here we first examine the original expression \(\left(2 x^{-3} y z^{-6}\right)(2 x)^{-5}\). In the first bracket, 2, \(x^{-3}\), y, and \(z^{-6}\) are all separate bases, each to the power of 1, -3, 1, and -6 respectively. In the second bracket, (2x) is the base and -5 is the exponent.

Step 2 - Distribute exponent in the second bracket

Next, distribute the exponent to both parts of the base. This is done by multiplying -5 to the exponents of the base. We get \(2^{-5} x^{-5}\).

Step 3 - Combine like bases

Then we combine the like bases by adding the exponents: two for 2 and two for x. Hence, we have \(2^{1-5} x^{-3 -5} y z^{-6}\). So far, we obtain: \(2^{-4} x^{-8} y z^{-6}\).

Step 4 - Interpret negative exponents

Note that any non-zero base with negative exponent implies a reciprocal of the base with a positive exponent. Thus, \(2^{-4}\) becomes \(1/2^{4}\), \(x^{-8}\) becomes \(1/x^{8}\), and \(z^{-6}\) becomes \(1/z^{6}\). Rewriting, we get: \(1/2^{4} \cdot 1/x^{8} \cdot y \cdot 1/z^{6}\).

Step 5 - Simplify

The last step is to simplify: \(1/2^{4}\) equals 1/16. So, we have \(1/16 x^{-8} y z^{-6}\) considering multiplication in mathematics is associative. It can be written as \(y/16x^{8}z^{6}\).