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Chapter 2: Problem 3

Simplify using the quotient rule. $$\frac{m^{7}}{m^{5}}$$

Short Answer

Expert verified
The simplified expression using the quotient rule for \(\frac{m^{7}}{m^{5}}\) is \(m^2\).

Step by step solution

01

Identify the base and exponents

In this expression, the base is m, and the exponents are 7 for the numerator and 5 for the denominator. The expression can be written as: \(\frac{m^7}{m^5}\).
02

Apply the quotient rule

According to the quotient rule for exponents, we subtract the exponent in the denominator from the exponent in the numerator when dividing with the same base. So, we will subtract 5 from 7, which results in 2: \(m^{(7-5)} = m^2\).
03

Simplified expression

Therefore, the simplified expression for the given problem is \(m^2\).

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental part of algebra that helps express repeated multiplication succinctly. When you see a number or variable with an exponent like \(m^7\), it means \(m\) is multiplied by itself 7 times.
This makes expressions shorter and much easier to work with, especially when dealing with large numbers or symbols like variables.
  • If the exponent is positive, as in our case \(m^7\), it indicates repeated multiplication.
  • When the exponent is negative, it suggests repeated division.
  • Zero exponents simplify any base to 1, meaning \(a^0 = 1\).
Understanding how exponents work is key to simplifying expressions, as they frequently appear in various algebraic problems.
Simplifying Expressions with the Quotient Rule
Simplifying expressions often involves rules of exponents, such as the quotient rule. The quotient rule is used when you divide two numbers or variables with the same base.
According to the quotient rule, you subtract the exponent in the denominator from the one in the numerator.
  • In our case, \(\frac{m^7}{m^5}\), both have the base \(m\).
  • Applying the rule, you subtract 5 (the exponent in the denominator) from 7 (the exponent in the numerator).
  • The result is \(m^{7-5} = m^2\).
This rule significantly simplifies expressions and makes calculations more manageable by reducing the power of the base, aiding further mathematical operations.
Algebra Essentials
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's a way to generalize arithmetic concepts and solve problems in a more abstract form.
To succeed in algebra, you must be comfortable with understanding and manipulating expressions.
  • An expression is a combination of numbers, variables, and operations like addition and multiplication.
  • Simplifying expressions involves reducing them to their simplest forms, often making them easier to work with.
  • Knowing rules such as the quotient rule for exponents is crucial for efficiently solving algebraic equations.
Algebra can be seen anywhere, from calculating interest rates to designing computer algorithms, demonstrating its importance in various fields.

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