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Chapter 5: Problem 25

Express using positive exponents and, if possible, simplify. $$\frac{z^{-4}}{3 x^{5}}$$

Short Answer

Expert verified
The simplified form is \(\frac{1}{3 x^{5} z^{4}}\).

Step by step solution

01

Rewrite Negative Exponent

Rewrite the expression so that all exponents are positive. Recall that a negative exponent can be converted by taking the reciprocal of the base. Therefore, rewrite the negative exponent \( \frac{z^{-4}}{3 x^{5}} \) as \(\frac{1}{z^{4}}\). Next, multiply this with the existing fraction: \(\frac{1}{3 x^{5} z^{4}}\).
02

Simplify Expression

The expression is already in its simplest form as there are no like terms or further factors to reduce. Therefore, the simplified expression is \(\frac{1}{3 x^{5} z^{4}}\).

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Key Concepts

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

Negative Exponents
Negative exponents can be confusing at first, but they follow a straightforward rule. A negative exponent indicates that you take the reciprocal of the base and then apply the positive exponent. For example, if you have an expression like \(x^{-n}\), it can be rewritten as \(\frac{1}{x^n}\). This rule simplifies calculations and helps to avoid the negative signs in exponents. In our example, we used this rule to turn \(z^{-4}\) into \(\frac{1}{z^4}\). This change makes the expression easier to work with and paves the way for further simplification.
Simplifying Expressions
Simplifying expressions is a crucial skill in algebra. It involves making an expression as simple as possible by combining like terms and reducing fractions. In our example: \(\frac{z^{-4}}{3 x^{5}}\), the first step was to rewrite the negative exponent as a positive exponent. After converting \(z^{-4}\) to \(\frac{1}{z^4}\), the expression then became \(\frac{1}{3 x^5 z^4}\). This final expression is simplified because there are no like terms to combine or common factors to reduce.
Algebraic Fractions
An algebraic fraction is a fraction in which the numerator or the denominator (or both) contain algebraic expressions. Handling algebraic fractions follows the same rules as regular numerical fractions, but it also involves special attention to the variables and exponents present. In our example, \(\frac{z^{-4}}{3 x^{5}}\) is an algebraic fraction. We handled it by applying the rules for negative exponents and simplifying the overall expression. Remember, whenever you deal with algebraic fractions, keep an eye out for negative exponents, common factors, and like terms to simplify.

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Most popular questions from this chapter

Review factoring expressions and solving equations. $$ x+5=0 $$

Find a value of the variable that shows that the two expressions are not equivalent. Answers may vary. $$ \frac{t^{6}}{t^{2}} ; t^{3} $$

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