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Chapter 10: Problem 49

Simplify. Write the answer with positive exponents only. $$\frac{x^{-2} y^{3}}{z^{-4}}$$

Short Answer

Expert verified
\(\frac{y^{3} z^{4}}{x^{2}}\)

Step by step solution

01

Address the Negative Exponents

Remember that for any nonzero number a and any integer n, the expression \(a^{-n} = \frac{1}{a^n}\). Rewrite the given expression using positive exponents: ewline ewline \(\frac{x^{-2} y^{3}}{z^{-4}}\) becomes \(\frac{y^{3}}{x^{2}} \times z^{4}\)
02

Simplify the Expression

Combine the terms to simplify the expression further: ewline ewline \(\frac{y^{3} \times z^{4}}{x^{2}}\)

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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 look tricky, but they simply mean that the number should be taken as a reciprocal. For any nonzero number \(a\) and any integer \(n\), the expression \(a^{-n} = \frac{1}{a^n}\). This rule helps to convert negative exponents into positive ones.
In our exercise, the expression \(x^{-2} \) means \( \frac{1}{x^2} \) and \( z^{-4} \) means \( \frac{1}{z^4} \). When simplifying expressions, we need to convert all negative exponents into positive by using this rule.
Positive Exponents
Positive exponents are straightforward鈥攖hey indicate how many times a number should be multiplied by itself. For instance, \(y^3\) means \(y \times y \times y\). In the exercise example, \(y^{3}\) remains the same because it's already positive. The goal is to make sure that all the exponents in our final answer are positive. After handling the negative exponents in the expression \( \frac{x^{-2} y^{3}}{z^{-4}} \), we rewrite it as \( \frac{y^{3}}{x^{2}} \times z^{4} \).
Fractions
Fractions indicate division and often consist of a numerator (top) and a denominator (bottom). In the given exercise, we start with a fraction \( \frac{x^{-2} y^{3}}{z^{-4}} \). Simplifying fractions with variables involves applying exponent rules. Converting the negative exponents gives us \( \frac{y^{3}}{x^{2}} \times z^{4} \). Now we combine terms properly to keep it in a simplified form.
Exponent Rules
Exponent rules are essential for simplifying expressions. Here are a few important ones:
  • \(a^{-n} = \frac{1}{a^n}\) converts negative exponents to positive.
  • \(a^m \times a^n = a^{m+n}\) helps when multiplying like bases.
  • \((a^m)^n = a^{mn}\) applies when raising powers to powers.
Using these rules, we see in the exercise: first, convert all negative exponents: \( \frac{x^{-2} y^{3}}{z^{-4}} \) to \( \frac{y^{3}}{x^{2}} \times z^{4} \). Then, multiply the terms: \( \frac{y^{3} z^{4}}{x^{2}} \). The result is fully simplified with only positive exponents.

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Most popular questions from this chapter

Perform the indicated operations. $$\left(-5 a^{2}-9 a^{3}+10 a^{4}\right)+\left(2 a^{3}-a^{4}\right)-\left(8 a+4 a^{3}+10 a^{4}\right)$$

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Which is greater? \(\left(4^{4}\right)^{5}\) or \(\left(4^{5}\right)^{4}\)
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