Question

\(\left(\frac{n^{2}-n^{3}}{n^{4}}\right)^{-2}\) is equivalent to which of the following? (A) \(\frac{n^{4}}{1-2 n+n^{2}}\) (B) \(\frac{n}{n^{8}}\) (C) \(\frac{n^{4}}{1-2 n+n^{2}}\) (D) \(\frac{n^{2}}{1+4 n-n^{2}}\)

Short answer

The expression is equivalent to (A) \( \frac{n^4}{1-2n+n^2} \).

Step-by-step solution

Simplify the expression inside the parentheses

The given expression is \( \left(\frac{n^{2}-n^{3}}{n^{4}}\right)^{-2} \). First, simplify \( \frac{n^{2}-n^{3}}{n^{4}} \). Factor \( n^2 \) out of the numerator: \( n^2(1-n) \). The expression becomes \( \frac{n^2(1-n)}{n^4} = \frac{1-n}{n^2} \).

Apply the negative exponent

Use the property \( (a)^m = \frac{1}{a^{-m}} \) to rewrite the expression with the negative exponent: \( \left(\frac{1-n}{n^2}\right)^{-2} = \left(\frac{n^2}{1-n}\right)^{2} \).

Square the expression

Square the expression from Step 2: \( \left(\frac{n^2}{1-n}\right)^{2} = \frac{n^4}{(1-n)^2} \).

Simplify the squared denominator

The denominator, \( (1-n)^2 \), can be expanded as follows: \((1-n)^2 = 1 - 2n + n^2\). Thus, the expression becomes \( \frac{n^4}{1 - 2n + n^2} \).

Identify the matching choice

The simplified expression is \( \frac{n^4}{1 - 2n + n^2} \), which matches option (A).

Key concepts

Exponent Properties

Understanding exponent properties is fundamental in solving many algebra problems, such as the one given. When you see an expression like \( (a)^m = \frac{1}{a^{-m}} \), it helps to transform complicated expressions and simplify them. Let's explore a few key properties:

• Product of Powers Property: When multiplying two powers with the same base, add the exponents. For example, \( x^a \times x^b = x^{a+b} \).
• Power of a Power Property: If you raise a power to another power, multiply the exponents: \((x^a)^b = x^{a \times b}\).
• Negative Exponent Property: A negative exponent signifies the inverse of the base raised to the absolute value of the power: \( x^{-a} = \frac{1}{x^a} \).

In the exercise, the negative exponent \( -2 \) is flipped to the positive through inversion. Mastery of these properties allows you to tackle expressions intuitively.

Rational Expressions

A rational expression is essentially a fraction but with polynomials in the numerator and/or the denominator. In our problem, this looks like \( \frac{n^2(1-n)}{n^4} \). The evaluation involves:

• Simplifying the Numerator and the Denominator: Take common factors out. Here, \( n^2(1-n) \) was simplified from the original numerator \( n^2 - n^3 \).
• Dividing the Expressions: By division, \( \frac{n^2(1-n)}{n^4} \) simplifies to \( \frac{1-n}{n^2} \).
• Flipping for Negative Exponents: In further steps, flipping due to negative exponents simplifies \( \frac{1-n}{n^2} \) to \( \frac{n^2}{1-n} \).

Understanding how to manipulate rational expressions equips you with the skills to work with complex algebraic fractions efficiently.

Polynomial Factorization

Polynomial factorization breaks polynomials into simpler components that, when multiplied together, give the original polynomial. It is a crucial step in simplifying the rational expressions seen in the exercise. For instance:

• Factoring Out Common Terms: To simplify \( n^2 - n^3 \), factoring out \( n^2 \) was necessary, resulting in \( n^2(1-n) \).
• Understanding Differences and Sums of Squares: While not directly applied in this problem, similar strategies involve recognizing patterns such as \( a^2 - b^2 = (a-b)(a+b) \).
• Revisiting the Squared Expressions: Expanding polynomials like \( (1-n)^2 \) was crucial. Here, it becomes \( 1 - 2n + n^2 \), supporting the need to know how to expand and simplify polynomial expressions.

Factorization not only simplifies problems but can reveal underlying structures, making problem-solving more intuitive.