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Chapter 0: Problem 110

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

Short Answer

Expert verified
So, the simplified form of the given exponential expression is \( \frac{x^{12} y}{27z^{7}} \)

Step by step solution

01

List out the given expression

The given expression is \( (3x^{-4}yz^{-7})(3x)^{-3} \). Each term has a base number and exponent. The bases include both numbers and variables.
02

Apply exponent rule to multiplication

After expanding the expression with the rule \( (ab)^n = a^n b^n \), we get \( 3^{-3}(x^{-4})^{-3}y(z^{-7}) \). The rule of exponents states that when we multiply two or more numbers or variables of the same base, we add their exponents together. Thus, it can also be viewed as \( 3^{-3}x^{-4*-3}yz^{-7} \).
03

Simplify the expression

Upon simplifying the expression we get: \( 3^{-3}x^{12}yz^{-7} \). Next, we look at the term \(3^{-3}\) which is negative because it's in the denominator. This follows the law of reciprocal where \(a^{-n}= 1/a^n\). Applying this rule gives us \(1/3^{3} , x^{12}, y, z^{-7}\).
04

Evaluate the numerical term

The numerical term \(1/3^{3}\) can be simplified to a numerical value. So, calculating, we get it as \(1/27\). So, our final expression now is \( 1/27 * x^{12} * y * z^{-7}\)

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