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Chapter 8: Problem 140

Combine like terms. $$ 7 m^{5}-2 m^{3}+8 m^{5}-m^{3} $$

Short Answer

Expert verified
15m^5 - 3m^3

Step by step solution

01

Identify like terms

Locate the terms that have the same variable and the same exponent. In this case, the like terms are: \(7m^5\) and \(8m^5\), also \(-2m^3\) and \(-m^3\).
02

Combine the terms with \(m^5\)

Add the coefficients of \(7m^5\) and \(8m^5\). This gives: \(7 + 8 = 15\), so \(7m^5 + 8m^5 = 15m^5\).
03

Combine the terms with \(m^3\)

Add the coefficients of \(-2m^3\) and \(-m^3\). This gives: \(-2 - 1 = -3\), so \(-2m^3 - m^3 = -3m^3\).
04

Write the final expression

Combine the results from the previous steps to get the final expression: \(15m^5 - 3m^3\).

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Key Concepts

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

Intermediate Algebra
Intermediate algebra focuses on consolidating the foundational knowledge from basic algebra and extending these concepts further. It often involves working with more complex equations and expressions. One of the critical skills learned in intermediate algebra is the ability to manipulate and simplify polynomial expressions effectively. This includes understanding how to combine like terms, factor expressions, and solve various types of equations. In this exercise, you practiced combining like terms, a fundamental step in simplifying algebraic expressions.
Polynomial Expressions
Polynomial expressions are mathematical expressions involving a sum of powers of variables with numerical coefficients. In simpler terms, they鈥檙e sums of terms that include variables raised to positive integer exponents and their coefficients. For example, expressions like \(7m^5 - 2m^3 + 8m^5 - m^3\) are polynomials. Polynomials can have multiple terms, and when simplifying them, it's essential to combine like terms, which are terms that have the same variable raised to the same power. This exercise required identifying and combining like terms such as \(7m^5\) and \(8m^5\), as well as \(-2m^3\) and \(-m^3\), leading to a simplified polynomial expression.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is one of the cornerstones of algebra. The goal is to rewrite expressions in the simplest and most efficient form. This generally involves combining like terms, factoring, and reducing fractions. In the given problem, simplifying the expression \(7m^5 - 2m^3 + 8m^5 - m^3\) starts with identifying like terms. Terms like \(7m^5\) and \(8m^5\) can be combined because they share the same base and exponent. Likewise, \(-2m^3\) and \(-m^3\) are combined. The process involved:
  • Locating terms with the same variable and exponent.
  • Adding or subtracting the numerical coefficients of these like terms.
  • Writing the simplified expression which, in this case, is \(15m^5 - 3m^3\).
It is essential to be methodical and precise to avoid mistakes and ensure accuracy when simplifying.

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Most popular questions from this chapter

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