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Magazine Chapter 5: Problem 108
Perform the indicated operations. a. \((m n)^{3}\) b. \((m+n)^{3}\)
Short Answer
Expert verified
a. \(m^3 n^3\); b. \(m^3 + 3m^2n + 3mn^2 + n^3\).
Step by step solution
01
Understanding the Expression
The first expression is \((mn)^3\), which means you need to cube the entire product of \(m\) and \(n\). This is equivalent to multiplying \((mn)\) by itself three times.
02
Applying the Power of a Product Rule
According to the power of a product rule, \((ab)^n = a^n b^n\). Therefore, \((mn)^3 = m^3 n^3\). You apply the exponent of 3 to both \(m\) and \(n\).
03
Simplification of Expression a
Multiply \(m^3\) and \(n^3\) to get the simplified expression: \(m^3 n^3\). This is the solution to exercise a.
04
Understanding the Expression
The second expression \((m+n)^3\) involves cubing the sum of \(m\) and \(n\). This requires expanding the expression by multiplying \((m+n)\) with itself three times.
05
Applying the Binomial Theorem
Use the binomial theorem to expand \((x+y)^n\), which is given by \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). For \((m+n)^3\), it expands to: \(m^3 + 3m^2n + 3mn^2 + n^3\).
06
Simplification of Expression b
The expanded form is \(m^3 + 3m^2n + 3mn^2 + n^3\). Each term is derived based on the coefficients and powers from the binomial expansion.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power of a Product
When you encounter an expression like \((mn)^3\), it is crucial to understand and properly apply the "Power of a Product" rule. This rule states that when you raise a product to a power, you must raise each factor in the product to that power independently.
- For example, in \((mn)^3\), both \(m\) and \(n\) are raised to the power of 3.
- This results in \((mn)^3 = m^3 \cdot n^3\).
Exponents
Exponents play a critical role in simplifying and understanding algebraic expressions. An exponent indicates the number of times a base is multiplied by itself. For example, in \(m^3\), the expression implies \(m\) is multiplied by itself twice more (three times in total).When combining the "Power of a Product" with exponents, you can see how potent these tools become in algebra. For instance:
- \((mn)^3\) using exponent rules translates into \(m^3 \times n^3\).
- This transformation utilizes the nature of exponents to distribute the power across each factor inside the parentheses.
Polynomial Expansion
Polynomial expansion is a technique utilized to expand expressions that consist of sums raised to a power鈥攁s in the example of \((m+n)^3\). This type of expansion usually involves the Binomial Theorem.The Binomial Theorem is a powerful tool that provides a formula to expand expressions like \((x+y)^n\) efficiently:
- For \((m+n)^3\), the expansion becomes: \[m^3 + 3m^2n + 3mn^2 + n^3\]
- Each component in the expanded form is derived from the original expression by calculating coefficients using combinations \(\binom{n}{k}\) and applying exponents to each variable.
