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Chapter 3: Problem 79

Simplify by combining like terms. $$-4 m^{5}-4 m^{5}$$

Short Answer

Expert verified
Question: Simplify the expression: $$-4 m^{5} - 4 m^{5}$$ Answer: $$-8 m^{5}$$

Step by step solution

01

Identify like terms

The given expression is: $$-4 m^{5} - 4 m^{5}$$ We can see that both terms are like terms, as they have the same variable (m), raised to the same power (5).
02

Combine like terms

To combine like terms, we add or subtract their coefficients. In this case, the coefficients are -4 and -4, so we add them together: $$(-4) + (-4) = -8$$
03

Write the simplified expression

Finally, we write the simplified expression with the combined coefficient and the variable with its power: $$-8 m^{5}$$ The simplified expression is: $$-8 m^{5}$$

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Key Concepts

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

Combining Like Terms
Every algebraic expression consists of terms, and keenly identifying like terms can make simplifying easier and faster. But what are like terms? These are terms that have the exact same variables raised to the same powers. For example, in the expression \(-4m^5-4m^5\), both terms share the variable \(m\) raised to the power 5, making them like terms.
  • Like terms can only be combined when they have the same variable and exponent.
  • This involves simply adding or subtracting the coefficients of these terms.
To dive into this concept, consider a simple analogy. Think of like terms as apples. You can add or subtract apples (\(4\) apples - \(2\) apples = \(2\) apples) but not mix them with pears.
Combining like terms simplifies the expression, making calculations and further algebraic operations more straightforward.
Coefficients in Algebra
Coefficients are like the stars of an algebraic term. In the expression \(-4m^5\), the \(-4\) is the coefficient, whereas \(m^5\) is the variable part.
  • Coefficients are the numbers directly in front of the variables in a term.
  • They represent how many times the variable combination is being counted.
Understanding coefficients helps when it comes to manipulating terms, especially when simplifying algebraic expressions.
When combining like terms, focus on the coefficients since the variable parts are identical. In our exercise, combine the coefficients \(-4\) and \(-4\) by simply adding them. Don't confuse this with more complex operations; the process is basic arithmetic.
Algebraic Simplification
Simplifying expressions in algebra is like tidying up; it makes work easier and results clearer. Simplification reduces expressions to their simplest form without changing their value. In expressions like \(-4m^5-4m^5\), it involves combining like terms as the first step.
Use simplification to:
  • Remove unnecessary complexity from expressions.
  • Make calculations more manageable.
  • Spot patterns or relationships within the expression.
A simplified expression is helpful when solving equations, as it reduces the chances of error and speeds up the process.
In summary, algebraic simplification uses concepts like combining like terms and understanding coefficients to bring clarity and efficiency to mathematical work.

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