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Chapter 10: Problem 376

In the following exercises, simplify. $$\frac{u^{9}}{u^{-2}}$$

Short Answer

Expert verified
The simplified expression is \(u^{11}\).

Step by step solution

01

Understand the Problem

The task is to simplify the expression \(\frac{u^9}{u^{-2}}\). This involves using the properties of exponents.
02

Apply the Quotient Rule

The Quotient Rule for exponents states that \(\frac{a^m}{a^n} = a^{m-n}\). Here, set \(a = u\), \(m = 9\), and \(n = -2\).
03

Perform the Subtraction of Exponents

Subtract the exponent in the denominator from the exponent in the numerator: \(u^{9 - (-2)} = u^{9 + 2}\).
04

Simplify the Exponent

Add the exponents: \(u^{9 + 2} = u^{11}\).
05

Write the Final Simplified Expression

The simplified form of the expression is \(u^{11}\).

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Key Concepts

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

Properties of Exponents
Exponents are a shorthand way of expressing repeated multiplication. For any number \(a\) raised to a power \(n\), written as \(a^n\), it means multiplying \(a\) by itself \(n\) times.

Several properties make working with exponents easier:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\). When multiplying like bases, add the exponents.
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\). When dividing like bases, subtract the exponents.
  • Power of a Power: \((a^m)^n = a^{m \times n}\). When raising an exponent to another power, multiply the exponents.
  • Negative Exponent: \(a^{-n} = \frac{1}{a^n}\). A negative exponent means taking the reciprocal.
Understanding and applying these properties correctly will help simplify expressions efficiently.
Quotient Rule for Exponents
One specific property of exponents is the Quotient Rule, which is particularly useful in division. The Quotient Rule states:
\(\frac{a^m}{a^n} = a^{m-n}\)

Here, \(a\) is the base, and \(m\) and \(n\) are the exponents in the numerator and denominator, respectively.

When we see an expression like \(\frac{u^9}{u^{-2}}\), we apply the quotient rule with \(a = u\), \(m = 9\), and \(n = -2\). This simplifies as follows:
  • Rewrite the expression using the quotient rule: \(u^{9-(-2)}\)
  • Subtraction of negative exponents is the same as adding: \(u^{9+2}\)

Thus, \(\frac{u^9}{u^{-2}}\) simplifies to \(u^{11}\).
This rule is key to handling more complex algebraic expressions with exponents.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves reducing them to their simplest form. This often means using properties of exponents efficiently. Consider our exercise example:
\(\frac{u^9}{u^{-2}}\)

Here's the detailed process:
  • Identify the rule applicable: The Quotient Rule applies here since we are dividing like bases.
  • Rewrite using the rule: \(u^{9 - (-2)}\)
  • Simplify the exponent: \(u^{9+2} = u^{11}\)
The expression is simplified from a fraction to a single exponent form: \(u^{11}\).

Simplifying expressions makes them easier to handle and reduces the complexity of algebraic problems.

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