Chapter 13: Problem 28
Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{x^{2}}{y}\right)^{-3}$$
Short Answer
Expert verified
\(\frac{y^3}{x^6}\)
Step by step solution
01
Understand the Negative Exponent Rule
Using the negative exponent rule, a negative exponent indicates that we should take the reciprocal of the base and then apply the positive exponent. For the expression \(\frac{x^{2}}{y}\)^{-3}, we would take the reciprocal of the fraction \(\frac{x^{2}}{y}\) and then cube it.
02
Take the Reciprocal of the Base
Before applying the exponent, we take the reciprocal of the base \(\frac{x^{2}}{y}\), which becomes \(\frac{y}{x^{2}}\).
03
Apply the Positive Exponent
Now, apply the cube (positive three exponent) to the reciprocal. Since the exponent applies to both the numerator and the denominator, \(\left(\frac{y}{x^{2}}\right)^3 = \frac{y^3}{(x^2)^3}\).
04
Simplify the Denominator
Next, simplify the exponent in the denominator by multiplying it. \((x^2)^3 = x^{2*3} = x^6\). So, our expression is now \(\frac{y^3}{x^6}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Expressions
Simplifying expressions in mathematics involves altering them into a more comprehensible or more standard form without changing their value. When simplifying, we follow established mathematical rules and properties to condense expressions into their simplest terms. This process often includes combining like terms, reducing fractions, or applying exponent rules, as in our example. Simplifying can make it easier to work with expressions, especially when solving equations or comparing values.
Considering the exercise \(\left(\frac{x^{2}}{y}\right)^{-3}\), we apply such principles by acknowledging the negative exponent, taking the reciprocal of the fraction, and then simplifying the result. The objective is to rewrite the expression in a form that is easier to understand and use in subsequent calculations. Simplification can be particularly helpful when dealing with complex algebraic expressions or interactive mathematical modeling.
Considering the exercise \(\left(\frac{x^{2}}{y}\right)^{-3}\), we apply such principles by acknowledging the negative exponent, taking the reciprocal of the fraction, and then simplifying the result. The objective is to rewrite the expression in a form that is easier to understand and use in subsequent calculations. Simplification can be particularly helpful when dealing with complex algebraic expressions or interactive mathematical modeling.
Reciprocal of a Fraction
The reciprocal of a fraction is simply a reversed version of the original fraction, obtained by swapping the numerator and the denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\) provided that neither \(a\) nor \(b\) is zero. It's the equivalent of dividing 1 by the fraction. The concept of reciprocals is crucial when working with division and fractions.
In the context of our example \(\left(\frac{x^{2}}{y}\right)^{-3}\), taking the reciprocal of the base \(\frac{x^{2}}{y}\) leads us to \(\frac{y}{x^{2}}\). The reciprocal is thus pivotal in simplifying expressions with negative exponents, as it transforms the initial operation into multiplication by the reciprocal of the base with a positive exponent.
In the context of our example \(\left(\frac{x^{2}}{y}\right)^{-3}\), taking the reciprocal of the base \(\frac{x^{2}}{y}\) leads us to \(\frac{y}{x^{2}}\). The reciprocal is thus pivotal in simplifying expressions with negative exponents, as it transforms the initial operation into multiplication by the reciprocal of the base with a positive exponent.
Exponent Rules
Understanding exponent rules, also known as the laws of exponents, is essential in algebra. These rules dictate how to handle expressions with exponents in various mathematical operations. Some fundamental rules include the product rule (multiplying with the same base), quotient rule (dividing with the same base), power rule (raising an exponent to another power), and the negative exponent rule, which is central to our discussion.
The negative exponent rule states that a number with a negative exponent is equal to the reciprocal of that number raised to the opposite positive exponent. Applying this rule to \(\left(\frac{x^{2}}{y}\right)^{-3}\), we first find the reciprocal of the fraction, then raise it to the positive third power. This approach converts a seemingly difficult problem into a series of simpler steps that utilize basic exponent rules, enhancing comprehension and making the process more manageable.
The negative exponent rule states that a number with a negative exponent is equal to the reciprocal of that number raised to the opposite positive exponent. Applying this rule to \(\left(\frac{x^{2}}{y}\right)^{-3}\), we first find the reciprocal of the fraction, then raise it to the positive third power. This approach converts a seemingly difficult problem into a series of simpler steps that utilize basic exponent rules, enhancing comprehension and making the process more manageable.
Mathematical Simplification
Mathematical simplification covers a variety of techniques used to streamline expressions or equations to their most basic form. Simplification often involves applying multiple mathematical rules sequentially. Through simplification, we aim to minimize complexity, thereby reducing the potential for errors and making it easier to identify the core components of an expression.
In our example, after applying the negative exponent rule \(\left(\frac{x^{2}}{y}\right)^{-3}\), we are left with \(\left(\frac{y}{x^{2}}\right)^3\), which then simplifies to \(\frac{y^3}{x^6}\). Simplifying \(x^{2}\) raised to the third power to \(x^{6}\) is an illustration of the power rule, which states that when raising a power to a power, you multiply the exponents. This final step completes the simplification process, yielding an expression where the negative exponent is resolved, and the expression is in its simplest form.
In our example, after applying the negative exponent rule \(\left(\frac{x^{2}}{y}\right)^{-3}\), we are left with \(\left(\frac{y}{x^{2}}\right)^3\), which then simplifies to \(\frac{y^3}{x^6}\). Simplifying \(x^{2}\) raised to the third power to \(x^{6}\) is an illustration of the power rule, which states that when raising a power to a power, you multiply the exponents. This final step completes the simplification process, yielding an expression where the negative exponent is resolved, and the expression is in its simplest form.
