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Chapter 0: Problem 112

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 expression simplifies to 1.

Step by step solution

01

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鈥檚 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 \(\frac {x^{-16} y^{-20} z^{-24}} {x^{16} y^{20} z^{24}}\)
02

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.
03

Final Simplification

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

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Most popular questions from this chapter

Evaluate each algebraic expression for the given value or values of the variable(s). $$x^{2}+3 x, \text { for } x=8$$

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\left\\{-9,-\frac{4}{5}, 0,0.25, \sqrt{3}, 9.2, \sqrt{100}\right\\}$$

$$\text { Factor completely.}$$ $$x^{4}-5 x^{2} y^{2}+4 y^{4}$$

This will help you prepare for the material covered in the next section. a. Use a calculator to approximate \(\sqrt{300}\) to two decimal places. b. Use a calculator to approximate \(10 \sqrt{3}\) to two decimal places. c. Based on your answers to parts (a) and (b), what can you conclude?

Give an example of a rational number that is not an integer.
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