Question

Rewrite the expression with positive exponents. $$x^{-2} y^{4}$$

Short answer

The expression \(x^{-2} y^{4}\) with positive exponents is \(y^{4}/x^2\).

Step-by-step solution

Identify the Base with Negative Exponent

The base with the negative exponent here is \(x\).

Apply the Negative Exponent Rule

The negative exponent rule tells us that any base \(a\) raised to the negative exponent \(-n\) is equal to \(1/a^n\). By applying this rule to our expression, we get that \(x^{-2}\) is equal to \(1/x^2\).

Rewrite the Original Expression with Positive Exponent

Now we can substitute \(x^{-2}\) with \(1/x^2\) in the original expression. Therefore, the expression \(x^{-2} y^{4}\) is rewritten with positive exponents as \(y^{4}/x^2\).

Key concepts

Exponent Rules

Understanding exponent rules is crucial when working with mathematical expressions involving powers. A fundamental aspect of these rules is dealing with negative exponents. The primary rule states: for any nonzero base a, raising it to a negative exponent -n is equivalent to the reciprocal of the base raised to the opposite positive exponent, which means \ta^{-n} = \( \frac{1}{a^n} \). This concept allows for the simplification of algebraic expressions and makes operations involving exponents more manageable.

In the exercise provided, we use this rule to transform \t\(x^{-2} \) into \t\(\frac{1}{x^2} \), making it easier to combine with other terms that may have positive exponents like \t\( y^{4} \).

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (like add, subtract, multiply, and divide). The beauty of algebraic expressions is that they can represent real-world situations in a concise and universally understood language. When simplifying expressions with negative exponents, the objective is often to represent them in standard form, which typically involves positive exponents.

For example, simplifying \t\(x^{-2} y^{4}\) to be \t\(\frac{y^4}{x^2} \) translates the expression into a more familiar and easily readable form, using positive powers only, which is in line with common algebraic conventions.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that helps make equations and expressions more straightforward and solvable. The process often involves applying exponent rules, combining like terms, and reducing fractions to their simplest form. In the context of negative exponents, simplification typically involves converting them to positive exponents to match the standard form expected in algebraic solutions.

For instance, transforming \t\(x^{-2}\) into \t\(\frac{1}{x^2}\) as seen in the exercise, paves the way for further simplification if additional terms or operations are involved in the expression. It's an example of using exponent rules to foster clarity and reduce potential for error in subsequent calculations.

Mathematical Notation

Mathematical notation is the language through which we communicate complex mathematical concepts succinctly and precisely. This system comprises symbols and numbers that denote operations, relationships, and quantities. In terms of negative exponents, proper notation is critical for clarity and accuracy.

The decision to express \t\(x^{-2} y^{4}\) as \t\(\frac{y^4}{x^2} \) demonstrates the value of consistent mathematical notation, as it adheres to the convention that prefers positive exponents in the final expression. Correct notation not only makes work easier to understand but also ensures that it can be correctly interpreted and utilized by others in further mathematical operations or applications.