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Chapter 3: Problem 1

Divide. $$x^{9} \div x^{2}$$

Short Answer

Expert verified
Answer: \(x^{7}\)

Step by step solution

01

Identify the base and exponents in the expression

In the expression, \(x^{9} \div x^{2}\), we can identify the base as x and the exponents as 9 and 2, respectively.
02

Subtract the exponents of the expressions

Since we have the same base, we can subtract the exponents to divide the expressions. We have: \(x^{9} \div x^{2} = x^{(9-2)}\).
03

Simplify the expression

Now we need to simplify the expression from step 2: \(x^{(9-2)} = x^{7}\). So, the final result is \(x^{7}\).

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Key Concepts

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

Understanding the Base
In any expression involving exponents, the base is a fundamental element. It's the number, variable, or term that is multiplied by itself.
For example, in the expression \(x^9\), the letter \(x\) is considered the base. The number above it, 9, is the exponent鈥攊t tells us how many times the base is used as a factor.

When dealing with algebraic problems, recognizing the base can help in manipulating and simplifying expressions easily.
  • Base tells us what is being multiplied.
  • In any exponential expression, the base remains consistent while the exponent may change.
Identifying the base is the first step in understanding and solving problems that involve exponents.
Dividing Exponents
Dividing exponents involves certain rules that make the process simple and intuitive. When you have the same base, you can divide the terms by subtracting the exponents.
For example, in the expression \(x^9 \div x^2\), both terms share the same base, \(x\). According to the rules of dividing exponents:
  • Keep the base the same.
  • Subtract the exponent of the denominator from the exponent of the numerator.
So, \(x^9 \div x^2\) becomes \(x^{(9-2)} = x^7\). This rule simplifies the process significantly. It's important to remember that this only applies when the bases in both the numerator and denominator are identical.
Simplify the Expression
After applying the rules for dividing exponents, the next step is to simplify the expression.
In this task, you've subtracted the exponents as follows: \(x^{(9-2)}\).
Now, calculate the subtraction: \(9 - 2 = 7\). Thus, the expression simplifies to \(x^7\).
  • Start by rewriting the expression with the updated exponent from the division process.
  • Double-check your calculations to ensure accuracy.
The goal of simplifying is to make the expression more manageable and to reveal the most straightforward version of it. Once simplified, it typically provides a clearer understanding of the problem's demands.

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