Problem 58 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. $$ \frac{9 x^{-3}}{y^{-1}} $$

Step-by-Step Solution

Answer

The expression \(\frac{9 x^{-3}}{y^{-1}}\) when rewritten with positive exponents is \(\frac{9y}{x^3}\)

Step 1: Handle \(x^{-3}\)

By the rule of exponents, we can re-write \(x^{-3}\) to get positive exponent by taking its reciprocal, i.e \(x^{-3} = \frac{1}{x^3}\). Hence, replacing \(x^{-3}\) in the original expression by its equivalent with positive exponent we get, \(\frac{9 \cdot \frac{1}{x^3}}{y^{-1}}\)

Step 2: Handle \(y^{-1}\)

Similarly, by the rule of exponents, \(y^{-1}\) can be re-written with positive exponent as \(\frac{1}{y}\). Substituting \(y^{-1}\) in the expression from Step 1, we get, \(\frac{9 \cdot \frac{1}{x^3}}{\frac{1}{y}}\)

Step 3: Simplify the Fraction

We now simplify this fraction to turn it into a simpler form by multiplying 9 by \(\frac{1}{x^3}\) in the numerator and multiplying the result by \(y\) in the denominator using the rule \( \frac{a}{\frac{b}{c}}= \frac{a*c}{b}\), which gives \( \frac{9 * \frac{1}{x^3}}{1} * y = \frac{9y}{x^3}\)

Key Concepts

Exponent Rules

The use of exponent rules is crucial in simplifying expressions with exponents, especially when dealing with negative ones. In this specific problem, we need to transform negative exponents into positive ones, applying rules of exponents, which state that any base raised to a negative power can be expressed as the reciprocal of the base raised to the positive power. For instance, in the expression \(x^{-3}\), this is equivalent to \(\frac{1}{x^3}\). Similarly, \(y^{-1}\) is re-written as \(\frac{1}{y}\). Knowing how to manipulate these expressions is foundational in understanding and solving algebraic problems. These transformations are not merely formal but central to making further calculations and simplifications possible.

Reciprocal

A reciprocal is a mathematical concept where you flip the numerator and the denominator of a fraction. In the context of exponents, a negative exponent implies taking the reciprocal. This means turning something like \(x^{-n}\) into \(\frac{1}{x^n}\). Effectively, the reciprocal is the flip of a fraction, which ties back to the exponent rule. It reflects the fundamental idea that raising a number to a negative power indicates that it belongs in a denominator, thus flipping its position in a fractional form. This is essential not only for rewriting expressions with positive exponents but also for interpreting and solving a wide range of mathematical problems. Whenever you're confronted with negative exponents, think of reciprocals as your helpful tool.

Simplifying Fractions

Simplifying fractions is a skill that allows expressions to be expressed in their simplest form. Once expressions are written with positive exponents, the next step is to simplify the resulting fraction. In this problem's context, simplifying involves applying the multiplication rule \(\frac{a}{\frac{b}{c}}= \frac{a \cdot c}{b}\). This means that when we have a fraction divided by another fraction, we multiply by the reciprocal of the denominator. In our exercise, \(\frac{9 \cdot \frac{1}{x^3}}{\frac{1}{y}}\) simplifies to \(\frac{9y}{x^3}\). This step not only reduces the expression to a more manageable form but also often prepares it for further operations, such as addition, subtraction, or integration with other mathematical elements. Simplifying fractions repeatedly employs the basic arithmetic operations of multiplication and division in strategic ways.