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Chapter 10: Problem 222

In the following exercises, simplify. $$\frac{u^{9}}{u^{3}}$$

Short Answer

Expert verified
The simplified expression is \(u^{6}\).

Step by step solution

01

Identify the Problem

The given exercise involves simplifying the expression \(\frac{u^{9}}{u^{3}}\).
02

Apply the Laws of Exponents

Use the quotient rule for exponents: when dividing like bases, subtract the exponent in the denominator from the exponent in the numerator. Thus, \(\frac{u^{9}}{u^{3}} = u^{9-3}\).
03

Perform the Subtraction

Subtract the exponents: \(9 - 3 = 6\). Therefore, \(\frac{u^{9}}{u^{3}} = u^{6}\).
04

Write the Simplified Expression

The simplified form of the given expression is \(u^{6}\).

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Key Concepts

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

Laws of Exponents
Exponents are a quick way to show repeated multiplication of the same number. Knowing how to handle them is essential in simplifying expressions. One essential principle is the 'laws of exponents'.

Some key laws include:
  • Product of Powers: When multiplying like bases, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
  • Power of a Power: When raising an exponent to another power, multiply the exponents. For example, \( (a^m)^n = a^{m \times n}\).
  • Quotient of Powers: When dividing like bases, subtract the exponents. For example, \( \frac{a^m}{a^n} = a^{m-n} \).


Understanding these laws of exponents helps you work with complex expressions more easily and to simplify them correctly.
Quotient Rule
The quotient rule is one of the essential laws of exponents and helps us simplify expressions that involve division. This rule states: when you divide two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

For example, in the exercise \( \frac{u^9}{u^3} \), you use the quotient rule because the base 'u' is the same in both the numerator and the denominator. The rule simplifies to: \( \frac{u^9}{u^3} = u^{9-3} \).

By mastering the quotient rule, you'll find it easier to handle and simplify many mathematical expressions and equations involving exponents.
Simplification Steps
To simplify an expression using the quotient rule, follow these straightforward steps:

  • Identify the Problem: Recognize that the expression involves exponents and needs simplification. For example: \( \frac{u^9}{u^3} \).
  • Apply the Quotient Rule: Use the rule which says to subtract the exponents: \( \frac{u^9}{u^3} = u^{9-3} \).
  • Perform the Subtraction: Calculate the difference between the exponents. Here, 9 - 3 = 6.
  • Write the Simplified Expression: The final result is expressed simply as \( u^6 \).


By following these steps, you break down the process and make it easier to manage. Each step helps simplify the task and ensures accuracy in your solution.

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Most popular questions from this chapter

Simplify. $$(14 t)^{0}$$

Multiply using any method. $$(4 y-1)\left(6 y^{2}-12 y+5\right)$$

Convert each number to decimal form. $$9.413 \times 10^{-5}$$

Simplify each expression. $$\frac{5^{6}}{5^{8}}$$

Simplify. $$\left(\frac{5 m}{n}\right)^{3}$$
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