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Chapter 9: Problem 73

Rewrite the expression using positive exponents. $$x^{-4} y^{3}$$

Short Answer

Expert verified
Hence, the expression \(x^{-4} y^{3}\) rewritten using only positive exponents is \(y^{3}/x^{4}\).

Step by step solution

01

Handle the Negative Exponent

The rules of exponents state that anything to the negative power is equal to one over the same thing to the positive power. So, \(x^{-4}\) can be re-written as \(1/x^{4}\).
02

Combine with Positive Exponent

Now, combining this with the remaining part of the expression which is \(y^{3}\), we rewrite the entire expression as \(1/x^{4} * y^{3}\).
03

Final Simplification

Although not necessary, it seems more suitable to arrange the expression to put the 'y' term first. Therefore, the simplified expression is \(y^{3}/x^{4}\). This becomes our final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponents
Negative exponents might seem intimidating at first, but they follow a simple rule. They essentially represent reciprocal powers. In the expression \(x^{-4}\), the negative exponent tells us to take the reciprocal of \(x\) and then raise it to the positive power. The key rule here is:
  • \(a^{-n} = \frac{1}{a^n}\)
This means \(x^{-4} = \frac{1}{x^4}\). This transformation helps turn seemingly complicated negative exponents into more manageable positive exponents. Understanding this rule is crucial in working with algebraic expressions that involve negative exponents.
Properties of Exponents
Exponents have several useful properties that allow us to manipulate them in different algebraic expressions:
  • Multiplying powers with the same base: \(a^m \cdot a^n = a^{m+n}\)
  • Dividing powers with the same base: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Power of a power: \((a^m)^n = a^{m\cdot n}\)
In our exercise, we used the property of rewriting a negative exponent to simplify our expression. Another property shown here is how multiplication can be expressed with division when a negative exponent is present. By recognizing these properties, we can easily manipulate even more complex expressions.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. In our example, the expression \(x^{-4} y^{3}\) involves both a rational and an integer exponent. Simplifying expressions with different types of exponents requires an understanding of the rules of exponents.
  • Terms: In \(x^{-4} y^{3}\), \(x^{-4}\) and \(y^{3}\) are the terms.
  • Simplification: To simplify \(x^{-4}\), we converted it to \(1/x^4\) and then multiplied it by \(y^{3}\) to obtain \(y^{3}/x^{4}\).
Mastering the ability to manipulate such expressions is essential in algebra. Understanding each component allows for a clearer pathway to simplify and solve various algebraic problems.

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