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Chapter 4: Problem 13

Find each product. $$ \left(6 m-\frac{4}{5} n\right)^{2} $$

Short Answer

Expert verified
36m^2 - \frac{48}{5} mn + \frac{16}{25} n^2

Step by step solution

01

Understand the equation

Recognize that the given expression \[ (6m - \frac{4}{5} n)^2 \] is a binomial squared. This can be expanded using the formula for the square of a binomial: \[ (a - b)^2 = a^2 - 2ab + b^2 \]
02

Identify the terms

In this expression, \( a = 6m \) and \( b = \frac{4}{5}n \).
03

Apply the formula

Substitute \( a \) and \( b \) into the binomial square formula: \[ (6m - \frac{4}{5} n)^2 = (6m)^2 - 2(6m)(\frac{4}{5} n) + (\frac{4}{5} n)^2 \]
04

Simplify each term

Calculate each part: \[ (6m)^2 = 36m^2 \] \[ 2(6m)(\frac{4}{5} n) = 2 \times 6m \times \frac{4}{5} n = \frac{48}{5} mn \] \[ (\frac{4}{5} n)^2 = \frac{16}{25} n^2 \]
05

Combine the simplified terms

Put all terms together: \[ (6m - \frac{4}{5} n)^2 = 36m^2 - \frac{48}{5} mn + \frac{16}{25} n^2 \]

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Key Concepts

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

Algebraic Expressions
An algebraic expression combines numbers, variables, and operators (such as +, -, *, and /). In this exercise, we see an expression made of two terms: \(6m\) and \(-\frac{4}{5}n\).
Expressions can represent mathematical relationships and are often simplified or expanded. Recognizing the structure allows us to apply the right formulas and methods.
Here, we use the algebraic expression inside the context of the binomial theorem.
Polynomials
Polynomials are a type of algebraic expression that consists of multiple terms combined using addition or subtraction. Each term includes a coefficient (a number) multiplied by a variable raised to a certain power.
For example, \(36m^2\), \(-\frac{48}{5} mn\) and \(\frac{16}{25} n^2\) are polynomial terms.
Polynomials are categorized by their degree, which is the highest power of the variable in the expression. In the solution, the degree of the polynomial is 2, as indicated by the term \(36m^2\).
Understanding polynomials can simplify complex problems and make it easier to manipulate algebraic expressions.
Binomial Theorem
The binomial theorem provides a quick way to expand expressions that are raised to a power. The basic formula for expanding \((a+b)^2\) is \((a+b)^2= a^2 + 2ab + b^2\).
In our exercise, we start with \((6m - \frac{4}{5}n)^2\). By identifying \(a = 6m\) and \(b = \frac{4}{5}n\), we can substitute these into the formula.
This gives us:
\((6m - \frac{4}{5}n)^2 = (6m)^2 - 2(6m)(\frac{4}{5}n) + (\frac{4}{5}n)^2\)
We then simplify each term:
\((6m)^2 = 36m^2\)
\(2(6m)(\frac{4}{5}n) = \frac{48}{5}mn\)
\((\frac{4}{5}n)^2 = \frac{16}{25}n^2\)
Finally, we combine these results to get the expanded form.
Using the binomial theorem allows us to handle complex expansions methodically.

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