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Chapter 5: Problem 132

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\left(\frac{x^{4} y^{5} z^{6}}{x^{-4} y^{-5} z^{-6}}\right)^{-4}$$

Short Answer

Expert verified
The simplified form of the given exponential expression is 1

Step by step solution

01

Apply the Power Rule

The Power Rule (a^(m/n) = (a^m)^n) lets us multiply exponents when they are inside a parenthesis and being raised to an exponent. Apply this rule to get \(x^{4*-4} y^{5*-4} z^{6*-4} / x^{-4*-4} y^{-5*-4} z^{-6*-4}\)
02

Multiply the Exponents

Perform the multiplications of the exponents to get \(x^{-16} y^{-20} z^{-24} / x^{16} y^{20} z^{24}\)
03

Simplify

Remember that \(a^{-m} = 1 / a^m\). Now, use this rule to change the negative exponents to positive by bringing them on the other side of the fraction thereby giving \(1 / x^{16} y^{20} z^{24} * x^{16} y^{20} z^{24}\)
04

Evaluate

Now, we can cancel out the equivalent terms on the top and bottom of the fraction, thereby giving a value of 1

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