Problem 57 - Algebra 1: Concepts and Skills

Question

Simplify. $$ \frac{2 m}{3} \cdot 6 m^{2} $$

Step-by-Step Solution

Answer

The simplified form of \(\frac{2 m}{3} \cdot 6 m^{2}\) is \(12m^{3}\).

Step 1: Multiply the Coefficients

Start by multiplying the fraction's numerator (the coefficient of the variable in the fraction) with the integer. Specifically, \(2 \times 6 = 12\).

Step 2: Add the Exponents

Next, multiply the variables which means you should add their exponents because both variables are the same (both are m). Since the first term \(m\) doesn't explicitly have an exponent, it's understood to be 1. Add the understood exponent of 1 to the actual exponent in the next term \(m^2\). Hence, the sum of the exponents is \(1 + 2 = 3\).

Step 3: Combine the Results

Combine the result from the two previous steps to form a single term. Multiply the result of the coefficients (12) with the result of the variable and its new exponent (\(m^3\)).

Key Concepts

Multiplying Coefficients

Simplifying algebraic expressions often begins with multiplying coefficients , which are the numerical parts of terms. When you are faced with a problem like \(\frac{2 m}{3} \cdot 6 m^{2} \), you need to focus on the numbers '2', '3', and '6'. Here’s what you do:

• Identify the coefficients in the expression. In our case \(\frac{2}{3}\) and '6' are the coefficients.
• Multiply these numbers together. You do this by multiplying 2 by 6, which equals 12.
• Remember that if you have coefficients in the form of a fraction, you need to multiply the top number (numerator) by the other coefficient just like any other number.

In this example, the fraction makes it a bit trickier, but you treat the numerator '2' just like you would if it were not part of a fraction. After multiplying with '6', you end up with the coefficient '12' for the simplified expression.

Exponent Rules

When working with algebraic expressions, understanding exponent rules is essential. Exponents tell us how many times to multiply a number or variable by itself. Here's a refresher on some basic exponent rules to remember:

• When you multiply like bases, you add the exponents (\(a^m \times a^n = a^{m+n}\)).
• An exponent of '1' is typically not written, so for \(m\), think of it as \(m^1\).
• If there's no exponent written, such as with \(m\) in our original problem, it's understood to be 1. Thus, \(m \cdot m^2 = m^{1+2} = m^3\).

In the exercise, \(m \cdot m^2\) is simplified by adding the exponents, resulting in \(m^3\). It's crucial to make sure that the bases you are combining are the same, otherwise, these rules do not apply.

Combining Like Terms

To combine like terms, look for terms in an expression that have the same variables raised to the same power. Here's how you can do this effectively:

• Scan the expression for terms that have matching variable parts. For instance, both may have \(m\), \(m^2\), or another similar variable component.
• Add or subtract the coefficients of these like terms just as you would with ordinary numbers.

In the context of our exercise, after applying the steps of multiplying coefficients and using exponent rules, you're left with '12' and \(m^3\). As these are not 'like terms', you can't combine them further, but you write them together as \(12m^3\). However, if we had terms such as \(12m^3 + 5m^3\), we would add the coefficients to end up with \(17m^3\), thus combining like terms.