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Chapter 10: Problem 541

Multiply the monomials. $$\left(\frac{1}{3} c^{2}\right)\left(30 c^{8}\right)$$

Short Answer

Expert verified
10c^{10}

Step by step solution

01

Identify the coefficients

Identify the numerical coefficients in each monomial. For \(\frac{1}{3} c^{2}\), the coefficient is \(\frac{1}{3}\). For \(30 c^{8}\), the coefficient is \(30\).
02

Multiply the coefficients

Multiply the coefficients together: \(\frac{1}{3} \times 30 = 10\).
03

Identify the variables and their exponents

Identify the variables and their exponents in each monomial. The first monomial is \(\frac{1}{3} c^{2}\), which has \(c^{2}\). The second monomial is \(30 c^{8}\), which has \(c^{8}\).
04

Apply the exponent rule

When multiplying like bases, add their exponents: \(c^{2} \times c^{8} = c^{2 + 8} = c^{10}\).
05

Combine the results from previous steps

Combine the results of the multiplied coefficients and the variables with their new exponent: \(10c^{10}\).

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Key Concepts

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

monomial multiplication
To multiply monomials, follow these steps. First, focus on the numerical parts, called coefficients. Then, look at the variable parts, considering their exponents. Multiplying monomials means handling both the coefficients and the variables, step by step.
Remember that a monomial is a single term consisting of a coefficient and a variable part. The variable part can include one or more variables raised to some power.
coefficients
Coefficients are the numerical parts of the monomials. For instance, in the monomial \(\frac{1}{3}c^{2}\), \(\frac{1}{3}\) is the coefficient. Similarly, in \(30c^{8}\), \(30\) is the coefficient.
In the original exercise, we need to multiply the coefficients first: \(\frac{1}{3} \times 30 = 10\). This gives us the numerical part of the final answer.
exponent rules
Exponent rules are crucial when multiplying monomials. The main rule is that when multiplying terms with the same base, add their exponents.
For example, in the original problem, we have \(c^{2} \times c^{8}\). Here, the base is 'c', and we add the exponents: \2 + 8\. This results in \(c^{10}\).
Always ensure that the bases are the same before applying this rule.
like bases
Like bases mean that the variables in the monomials are the same. In our example, both terms have the variable 'c'. This allows us to use the exponent addition rule.
If the bases were different (for example, \(c^{2} \times d^{8}\)), we would handle each variable separately and not combine the exponents.
In summary, only combine exponents if the variables (bases) are the same.

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Most popular questions from this chapter

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