Problem 12 Determine whether the expression... [FREE SOLUTION]

Chapter 3: Problem 12

Determine whether the expression is a monomial. Explain. $$5 m^{3} n^{7} p$$

Short Answer

Expert verified

Explain your reasoning. Answer: Yes, the expression $$5m^3n^7p$$ is a monomial because it consists of a single term with a real number coefficient (5) and variables (m, n, and p) raised to non-negative integer powers (3, 7, and 1).

Step by step solution

Identify the coefficient and variables

In the given expression, $$5m^3n^7p$$, the coefficient is the real number 5, and the variables are m, n, and p.

Check the powers of the variables

Next, we need to check if the variables' powers are non-negative integers. In this expression, we have: $$m^3$$, $$n^7$$, and $$p^1$$ (since p has an implied power of 1). The powers of all variables are non-negative integers (3, 7, and 1).

Confirm that the expression is a monomial

Since the expression consists of a single term with a real number coefficient and variables raised to non-negative integer powers, we can confirm that $$5m^3n^7p$$ is a monomial. In conclusion, the expression $$5m^3n^7p$$ is a monomial because it meets the criteria of a monomial, having a real number as its coefficient and variables raised to non-negative integer powers.

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Key Concepts

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

Coefficient

In algebra, a coefficient is a real number or constant that multiplies the variables in an expression. For the expression $5m^3n^7p$, the number 5 is the coefficient. It represents how many times the product of the variables is counted or multiplied. Understanding coefficients is essential because:

It determines the magnitude or size of the term in an expression.
A coefficient can be positive, negative, or zero.
It helps us identify terms within longer expressions or equations.

In the context of a monomial, having a real number as a coefficient is part of what qualifies the expression as such. This means any whole, fractional, or decimal number can be a coefficient, as long as it is real.

Variables

Variables are symbols that represent unknown values or quantities in mathematical expressions. In the monomial $5m^3n^7p$, the letters $m$, $n$, and $p$ are variables.Variables can take on different values, which is why they are essential in forming expressions and equations. Here are some key points about variables:

They allow us to write general formulas that can represent multiple scenarios.
Variables can be raised to powers to indicate repeated multiplication.
In expressions, variables are critical as they are where manipulation and solving occur.

In expressions that are monomials, like the one we are examining, variables appear raised to non-negative integer powers. It is this unique structural organization that helps identify a monomial.

Integer Powers

When we talk about integer powers in algebra, we're referring to the exponent of a variable. An integer power indicates how many times a variable is multiplied by itself. For example, in the expression $5m^3n^7p$, $m$ is raised to the third power, indicated as $m^3$. This means $m$ is multiplied by itself three times.Understanding powers is crucial because:

It helps us to execute operations like multiplication and division more efficiently.
Integer powers simplify expressions by condensing repeated multiplication into a compact form.
In a monomial, all powers must be non-negative integers (0, 1, 2, etc.).

Integer powers must always be non-negative for it to be part of a monomial. This means variables cannot have negative or fractional exponents in a monomial.

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Determine whether the expression is a monomial. Explain. $$5 m^{3} n^{7} p$$

Short Answer

Step by step solution

Identify the coefficient and variables

Check the powers of the variables

Confirm that the expression is a monomial

Key Concepts

Coefficient

Variables

Integer Powers

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