Problem 63 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \left(r^{2}+10 s\right)^{2} $$

Step-by-Step Solution

Answer

The expanded expression is \(r^{4} + 20sr^{2} + 100s^{2}\).

Step 1: Identify the Expression Type

The expression \((r^{2}+10s)^{2}\) is a binomial square. This means we must apply the formula \((a+b)^{2}=a^{2}+2ab+b^{2}\) to expand it.

Step 2: Apply the Binomial Square Formula

First, identify \(a\) and \(b\) in the expression. Here, \(a = r^2\) and \(b = 10s\). Use the formula: \[(r^{2}+10s)^{2} = (r^{2})^{2} + 2(r^{2})(10s) + (10s)^{2}\]

Step 3: Calculate Each Term

Now calculate each term separately: - The first term is \((r^{2})^{2} = r^{4}\).- The second term is \(2 imes r^{2} imes 10s = 20sr^{2}\).- The third term is \((10s)^{2} = 100s^{2}\).

Step 4: Combine the Terms

Combine all terms calculated in Step 3 to get the expanded form of the expression: \[r^{4} + 20sr^{2} + 100s^{2}\]

Key Concepts

Binomial Square

A binomial square is an important concept in algebra. Essentially, it refers to squaring a binomial expression, which includes two terms. The standard form of a binomial is \((a+b)\) or \((a-b)\). When you square a binomial, like \((a+b)^2\), it expands into a trinomial through a simple formula:

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)

In this exercise, we work with the binomial \((r^2 + 10s)\). Squaring this involves applying the formula: it helps us expand and simplify the expression in a structured way.
Simplifying binomials is fundamental in algebra and calculus as it aids in solving complex equations and can make them more approachable.

Expression Expansion

Expression expansion involves breaking down a complex expression into a simpler form. Here, expanding the binomial square \((r^2 + 10s)^2\) involves identifying the components:

• Firstly, recognize \(a = r^2\) and \(b = 10s\).
• Apply the expansion formula \((a+b)^2 = a^2 + 2ab + b^2\).

This approach makes each term clear and simplified:

• \((r^2)^2\) becomes \(r^4\).
• \(2(r^2)(10s)\) simplifies to \(20sr^2\).
• \((10s)^2\) becomes \(100s^2\).

The expanded form, \(r^4 + 20sr^2 + 100s^2\), clarifies each term's origin and contribution to the final expression. This procedural expansion is essential for managing algebraic equations effectively.

Step-by-Step Solution

Breaking down the problem into manageable parts is essential for clarity. The step-by-step solution method allows you to see every part of the calculation separately:

• Step 1 begins with recognizing our target expression as a binomial square \((r^2+10s)^2\).
• Step 2 involves deciding the formula to use and identifying \(a\) and \(b\) for our expressions \(a = r^2, b = 10s\).
• Step 3 involves individually calculating each part of the formula:


• Calculate \((r^2)^2\) giving \(r^4\).
• Compute \(2 \, \times \, r^2 \, \times \, 10s\) simplifying to \(20sr^2\).
• Find \((10s)^2\) yielding \(100s^2\).
• Finally, in Step 4, all calculated terms are combined into \(r^4 + 20sr^2 + 100s^2\).

This process is crucial for solving algebraic problems efficiently. It helps track intricate steps and reduces errors.