Question

Evaluate. $$\frac{\left(z^{-n}\right)\left(z^{2}\right)}{z^{2-n}}$$

Short answer

The simplified expression is 1, assuming that \(z\) is not equal to zero.

Step-by-step solution

Simplify the Numerator

Multiply the terms in the numerator using the law of exponents, which states that when multiplying two powers that have the same base, you add the exponents. In this case, you have \(z^{-n}\) and \(z^2\). The base is \(z\) and the exponents are \(-n\) and \(2\). So, you add \(-n + 2\) to get \(z^{2-n}\).

Dividing Powers With the Same Base

Now that we have the same expression \(z^{2-n}\) in both the numerator and the denominator, we can simplify the expression. Since the expressions in the numerator and the denominator are identical, \(z^{2-n} / z^{2-n} = 1\), assuming that \(z\) is not equal to zero.

Write the Final Answer

After simplifying, the entire expression evaluates to \(1\), given that the base \(z\) is not equal to zero.

Key concepts

Simplifying Expressions

Understanding how to simplify algebraic expressions is a fundamental skill in algebra. An expression is considered simplified when it is reduced to the simplest form possible. This process often involves combining like terms, which are terms that have the same variable raised to the same power, and performing operations that make the expression more concise.

For example, when simplifying the expression \( \frac{(z^{-n})(z^{2})}{z^{2-n}} \), we begin by recognizing the like base \(z\) in all components. Simplification becomes easier when we adhere to the hierarchy of operations and the laws of exponents to manage the individual parts of the expression before attempting to simplify the overall equation.

Exponent Rules

The laws of exponents are crucial tools for working with algebraic expressions involving powers. Here's a quick rundown of these rules:

• \textbf{Product of Powers}: When multiplying two exponents with the same base, you add the exponents, e.g., \( a^m \cdot a^n = a^{m+n} \).
• \textbf{Quotient of Powers}: When dividing two exponents with the same base, you subtract the exponents, e.g., \( \frac{a^m}{a^n} = a^{m-n} \).
• \textbf{Power of a Power}: To raise an exponent to another power, you multiply the exponents, e.g., \( (a^m)^n = a^{mn} \).
• \textbf{Negative Exponent}: A negative exponent represents the reciprocal of the base raised to the opposite positive exponent, e.g., \( a^{-n} = \frac{1}{a^n} \).
• \textbf{Zero Exponent}: Any non-zero base raised to the zero power is equal to one, e.g., \( a^0 = 1 \).

These rules allow us to tackle the given exercise methodically. In our example, the product and quotient of powers are used to simplify the expression. Recognizing these exponent rules and applying them correctly is key to mastering algebraic manipulation.

Algebraic Manipulation

Algebraic manipulation involves rearranging and reworking equations and expressions to solve for variables or simplify the expression. In the exercise \( \frac{(z^{-n})(z^{2})}{z^{2-n}} \), the manipulation starts with using the exponent rules to combine powers in the numerator and then comparing the result with the denominator.

Through this manipulation, we identify that the numerator \(z^{-n}\cdot z^{2} = z^{2-n} \) matches the denominator, leading us to a simple conclusion that \( \frac{z^{2-n}}{z^{2-n}} = 1 \) as long as \(z \eq 0\). Effective algebraic manipulation is about breaking down complex expressions into simpler parts and reassembling them to showcase the inherent simplicity of algebra.