Problem 68 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \left(q^{6}+\frac{1}{3}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(q^{12} + \frac{2}{3}q^6 + \frac{1}{9}\).

Step 1: Identify the Binomial

The expression is a binomial because it consists of two terms: \(q^6\) and \(\frac{1}{3}\). We need to square this binomial.

Step 2: Understand the Square of a Binomial Formula

The formula for squaring a binomial \((a + b)^2 = a^2 + 2ab + b^2\). In this problem, \(a = q^6\) and \(b = \frac{1}{3}\).

Step 3: Apply the Formula for Squaring a Binomial

Substitute \(a = q^6\) and \(b = \frac{1}{3}\) into the formula: \[(a + b)^2 = (q^6)^2 + 2(q^6)(\frac{1}{3}) + (\frac{1}{3})^2\]

Step 4: Calculate the Square of Each Term

Calculate \((q^6)^2\) which equals \(q^{12}\), and \((\frac{1}{3})^2\) which equals \(\frac{1}{9}\). Thus, we have:\[q^{12} + 2(q^6)(\frac{1}{3}) + \frac{1}{9}\]

Step 5: Multiply the Middle Term

Calculate the product \(2(q^6)(\frac{1}{3})\):\[2 \cdot q^6 \cdot \frac{1}{3} = \frac{2}{3}q^6\].

Step 6: Simplify the Expression

Combine all terms: \[q^{12} + \frac{2}{3}q^6 + \frac{1}{9}\] is the simplified expression.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra. It allows us to expand expressions raised to powers without directly multiplying each term. When dealing with an expression like \[(a + b)^n\],the Binomial Theorem gives us a formula to find the expanded form. The theorem states:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\].For squaring a binomial, this simplifies to a special case \((a + b)^2 = a^2 + 2ab + b^2\).In our exercise, **\(a = q^6\)** and **\(b = \frac{1}{3}\)**. We apply this directly to find the expanded expression. It's all about recognizing the structure and applying the formula! Practice makes perfect with these types of algebraic operations.

Polynomial Operations

Polynomial operations, such as addition, subtraction, and multiplication, are fundamental in algebra. When operating with polynomials, you often combine and simplify terms. For instance:

• **Addition and Subtraction:** Simply combine like terms. These are terms that have the same variables raised to the same powers.
• **Multiplication:** Apply the distributive property, or use formulas like the **square of a binomial** to simplify the process.

In our problem, we squared a binomial using the formula. Each term of the binomial is treated as a separate polynomial term. We calculated individually for each term and then combined them together to simplify the expression. Understanding how to operate with polynomials is essential for handling more complex algebraic expressions and equations.

Algebraic Expressions

An algebraic expression consists of variables, constants, and arithmetic operations. Understanding this concept is crucial as it forms the basis of algebra. In our case:

• **Variables:** This is usually represented by letters like \(q\) in our exercise.
• **Constants:** Numbers that are not variables, like \(\frac{1}{3}\).
• **Operations:** These include addition, subtraction, multiplication, etc.

For the expression \( (q^6 + \frac{1}{3})^2 \), each part plays a role. The algebraic operation of squaring takes place based on the binomial structure. Recognizing how each component of an expression affects the whole allows for successful manipulation and simplification, leading to correct solutions. Mastery of algebraic expressions is vital for solving equations and understanding math in depth.