Problem 27 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$ \left(b^{3} c^{4}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(b^6 c^8\).

Step 1: Apply the Power of a Power Rule

When raising a power to another power, multiply the exponents. Therefore, for each component inside the parentheses, apply this rule: \[\left(b^{3} c^{4}\right)^{2} = b^{3 \times 2} \times c^{4 \times 2} = b^{6} c^{8}\]

Step 2: Write the Final Expression with Positive Exponents

Ensure that all the exponents in the final expression are positive. The expression \(b^{6} c^{8}\) already has positive exponents, so no further action is needed.

Key Concepts

Power of a Power Rule

The "Power of a Power Rule" in exponentiation is essential when dealing with nested exponents. It helps us simplify expressions where one power is raised to another power. The core idea behind this rule is straightforward: when you have an expression like \((a^{m})^{n}\), you multiply the exponents \((m imes n)\).
For example, let's apply this rule to the expression \((b^3 c^4)^2\):

• Focus separately on each base within the parentheses.
• Multiply their respective exponents by the outside exponent (2 in this case).

So it becomes: \([b^{3 \times 2} \cdot c^{4 \times 2}]\). This results in \([b^6 \cdot c^8]\).
Notice how the calculation is direct and systematic. By understanding and applying this rule effectively, you can simplify complex expressions quickly.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while retaining the same value.
This task may include applying mathematical rules, such as combining like terms or using exponent rules. With exponential expressions like our given one \((b^3 c^4)^2\), simplification focuses on reducing the power products without changing the bases, ensuring all terms are clear and concise.
Here's how you simplify:

• Apply the Power of a Power rule to manage the exponents first.
• Reorganize the expression with only positive exponents where necessary.

The result, in this case, becomes \([b^6 \cdot c^8]\). Remember, keeping expressions simple is vital for making calculations easier, particularly for more complicated mathematical problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are the building blocks of algebra.
These expressions let us express mathematical relationships and patterns in a concise form. When dealing with algebraic expressions such as \((b^3 c^4)^2\), it's important to understand the role each element plays.
In algebra:

• **Variables** (like \(b\) and \(c\)) represent unknown or varying values
• **Exponents** indicate how many times a variable or number is multiplied by itself.

The expression \((b^6 \cdot c^8)\) is simplified but still represents the powerful notion of how variables interact under multiplication and exponentiation. Recognizing how to interpret and manipulate these expressions is crucial for solving broader mathematical challenges.