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Chapter 5: Problem 22

Express using positive exponents and, if possible, simplify. $$x y^{-9}$$

Short Answer

Expert verified
\( \frac{x}{y^9} \)

Step by step solution

01

Identify Negative Exponent

The given expression is \(x y^{-9}\). Notice that the exponent of \(y\) is -9, which is a negative exponent.
02

Rewrite Using Positive Exponent

To express \(y^{-9}\) with a positive exponent, use the property \(a^{-n} = \frac{1}{a^n}\). Therefore, \(y^{-9} = \frac{1}{y^9}\).
03

Combine Terms

Replace \(y^{-9}\) in the original expression \( x y^{-9} \) with \( \frac{1}{y^9} \). This gives \( x \cdot \frac{1}{y^9}\).
04

Simplify the Expression

Simplify the expression \( x \cdot \frac{1}{y^9}\) to obtain \( \frac{x}{y^9}\).

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Key Concepts

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

Exponent Rules
Understanding exponent rules is crucial for simplifying expressions with exponents. Exponents are a shorthand way to represent repeated multiplication of the same number. For example, the expression \(a^n\) means that 'a' is multiplied by itself 'n' times. Here are some basic rules:
  • Product of powers rule: \(a^m \times a^n = a^{m+n}\)
  • Quotient of powers rule: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Power of a power rule: \((a^m)^n = a^{m \times n}\)
  • Negative exponent rule: \(a^{-n} = \frac{1}{a^n}\)
These rules help us manipulate and simplify expressions with exponents.
Negative Exponents
Negative exponents might seem tricky, but they're just as manageable as positive ones. A negative exponent means that the base is on the wrong side of a fraction. So, instead of multiplying, you're actually dealing with division. The rule \(a^{-n} = \frac{1}{a^n}\) tells us that any number raised to a negative exponent is the same as one divided by that number raised to the positive exponent.
For example:
  • \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)
  • \(x^{-1} = \frac{1}{x}\)
By converting negative exponents into positive ones, we can simplify expressions more easily. In our exercise, converting \(y^{-9}\) to \(\frac{1}{y^9}\) helps us move further along with the solution.
Simplifying Expressions
Simplifying expressions involves reducing them to their most basic form, so they're easier to understand and work with. Using the exponent rules and working step by step can make this easier.
For our specific example:
  • Start with the expression: \(x y^{-9}\)
  • Convert the negative exponent: \(y^{-9} = \frac{1}{y^9}\)
  • Combine terms: \(x y^{-9} = x \cdot \frac{1}{y^9}\)
  • Simplify to obtain: \(\frac{x}{y^9}\)
By converting negative exponents and following the proper rules, you can simplify any given expression efficiently. This makes it not only easier to understand but also simpler to work with in further mathematical problems.

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