91Ó°ÊÓ

Combining like terms is a fundamental concept in algebra. It helps simplify expressions and makes solving equations easier.
Like terms are terms that have the same variable raised to the same power. For example, in the expression \(3x^2 + 5x^2\), both terms are like terms because they both have the variable \(x\) raised to the power of 2. These can be combined by adding their coefficients:
\[ 3x^2 + 5x^2 = 8x^2 \]
In the exercise, similar steps are applied when combining terms with the same variable bases: \(m\) and \(n\). We focus on merging terms that share the same base and add their exponents to get a simpler, combined term.

Exponent rules govern how we handle exponents in mathematical expressions. These rules are crucial for multiplying polynomials.
The most relevant exponent rule for this exercise is the Product of Powers Rule. This rule states: \[ a^m \times a^n = a^{m+n} \]
In simpler terms, when you multiply two expressions with the same base, you add their exponents. Here's how we use the rule in the problem:
For \(m\): \[ m \times m^4 = m^{1+4} = m^5 \]
Similarly, for \(n\): \[ n^3 \times n^1 = n^{3+1} = n^4 \]
By applying this rule, we significantly simplify the process of handling exponents during multiplication.

The distributive property is a key concept in algebra that allows us to simplify expressions and solve equations more effectively. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. The property is written as: \[ a(b+c) = ab + ac \]
In this problem, we apply the distributive property in a slightly different way. Instead of distributing over addition or subtraction, we distribute over multiplication:
\[ 6mn^3(3m^4n) \]
This means we multiply each part of the first polynomial by each part of the second polynomial, starting with the constants: \[ 6 \times 3 = 18 \]
Then we multiply the variables separately using exponent rules as discussed earlier. Putting it all together:
\[ 6mn^3 \times 3m^4n = 18m^5n^4 \]
The distributive property essentially breaks down the multiplication process, making it easier to handle larger expressions.