Problem 92 - Algebra 2

Question

Which expression is equivalent to \(\left(n^{\frac{3}{2}} \div n^{-\frac{1}{6}}\right)^{-3} ?\) \(\begin{array}{llll}{\text { A. } n^{27}} & {\text { B. } n^{-27}} & {\text { C. } n^{-4}} & {\text { D. } n^{-5}}\end{array}\)

Step-by-Step Solution

Answer

D. \(n^{-5}\)

Step 1: Simplify the Expression Inside the Parenthesis

First deal with the division inside the brackets; we subtract the exponents when we divide terms with like bases. Hence, \(\left(n^{\frac{3}{2}} \divn^{-\frac{1}{6}}\right) = n^{\frac{3}{2} - (-\frac{1}{6})} = n^{\frac{3}{2} + \frac{1}{6}} = n^{\frac{9}{6} + \frac{1}{6}} = n^{10/6} = n^{\frac{5}{3}}. \)

Step 2: Applying the Rule of Exponents

Now, we have to deal with the power that applies to the whole bracket, which is -3. So now our expression becomes: \(\left(n^{\frac{5}{3}}\right)^{-3}\). In this case, we need to multiply the exponents as this is the law for powers of powers in exponents: \(a^{mn} = (a^m)^n \)So, \(\left(n^{\frac{5}{3}}\right)^{-3} = n^{\frac{5}{3} * -3} = n^{-5}\).

Step 3: Final Answer

Comparing with the given options, \(n^{-5}\) matches with option D. So, D is the correct answer.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power or exponent. This particular process symbolizes repeated multiplication of the base. For instance, in the expression \(a^n\), the base \(a\) is multiplied by itself \(n\) times. If \(n\) is a positive whole number, this is straightforward. For example, \(2^3 = 2 \times 2 \times 2 = 8\).
Exponentiation forms the bedrock for numerous complex calculations in mathematics. It is vital to understand that the operations follow specific laws, such as the power of a product or quotient. Mastery of these laws facilitates easy handling of expressions involving powers and exponents.
Remember, exponents can be fractions, which resemble roots. For instance, \(a^{\frac{1}{2}}\) is equivalent to the square root of \(a\). Certain operations involve negative numbers in the exponent and will be discussed later.

Arithmetic Operations with Exponents

It is fundamental to comprehend how different arithmetic operations interact with exponents. Generally, three main rules simplify these operations:

• When multiplying expressions with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• When dividing, subtract the exponents: \(a^m \div a^n = a^{m-n}\).
• For powers of powers, multiply the exponents: \((a^m)^n = a^{m \times n}\).

These rules allow us to solve complex problems by breaking them down into simpler steps. For instance, in the given exercise, the expression inside the parentheses was divided, which required subtracting the exponents.
This logic helps streamline calculations, making it easier to obtain a solution efficiently. Applying these rules systematically is crucial to solving expressions involving exponents correctly.

Negative Exponents

Negative exponents represent a reciprocal operation in mathematics. Instead of multiplying the base, dividing is implied. Hence, \(a^{-n}\) equates to \(\frac{1}{a^n}\). This bypasses the need for traditional division because it transforms exponentiation into an inverse form.
Understanding negative exponents is crucial, especially when dealing with complex expressions like the one in the exercise. With an expression inside a parenthesis raised to a negative power, one must remember to take the reciprocal to simplify it effectively.
In this exercise, the final result \(n^{-5}\) shows the reciprocal of \(n^5\). This transformation is critical in ensuring that expressions are both simplified and solved correctly. Mastering this concept enables effective handling of a wide range of mathematical problems, embedding proficiency in negative exponents' manipulation.