Problem 112 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Apply Power of Power Rule

Apply the power of a power rule to each term inside the parentheses. This will involve multiplying the exponents of each term with the exponent that’s outside the parentheses. Because of the negative exponent, it'll reverse the sign of each of the exponents.\( \left(\frac {x^{4* -4}y^{5*-4} z^{6 * -4}} {x^{-4*-4}y^{-5 *-4}z^{-6*-4}} \right)\) is then equal to \(\frac {x^{-16} y^{-20} z^{-24}} {x^{16} y^{20} z^{24}}\)

Step 2: Simplify the Expression

Now apply the rule which states that \(a^n / a^n = 1\) which simplifies representation to \( \frac {x^{-16} / x^{16}} {y^{-20} / y^{20}} {z^{-24}/ z^{24}} \) , which results in 1.

Step 3: Final Simplification

Since the expression now equals 1, there are no more steps to take. The expression is simplified to its smallest form.