Problem 9 - Elementary Algebra

Question

Make use of either or both the power rule for products and the power rule for powers to simplify each expression. $$ \left[\left(12 c^{4} u^{3}(w-3)^{2}\right]^{5}\right. $$

Step-by-Step Solution

Answer

Question: Simplify the following expression using the power rule for products and the power rule for powers: $$ \left[\left(12 c^{4} u^{3}(w-3)^{2}\right]^5 \right. $$ Answer: The simplified expression is: $$ 248832c^{20}u^{15}(w-3)^{10} $$

Step 1: Distribute power of 5 to each term

Start by raising each term inside the parentheses to the power of 5: $$ (12^5)(c^5)^{4}(u^5)^{3}((w-3)^5)^2 $$

Step 2: Apply the power rule for powers

Now, we will use the power rule for powers, which states that when raising a power to another power, we multiply the exponents: $$ (12^5)(c^{5 \cdot 4})(u^{5 \cdot 3})((w-3)^{5 \cdot 2}) $$

Step 3: Simplify exponents

Calculate the new exponents: $$ (12^5)(c^{20})(u^{15})((w-3)^{10}) $$

Step 4: Calculate 12 raised to the power of 5

Calculate \(12^5\): $$ 248832(c^{20})(u^{15})((w-3)^{10}) $$

Step 5: Write the final simplified expression

The simplified expression is: $$ 248832c^{20}u^{15}(w-3)^{10} $$

Key Concepts

Power Rule for Powers

The Power Rule for Powers is an important principle in mathematics that helps simplify expressions where an exponent is raised to another exponent. This rule makes calculations simpler and reduces complex expressions into more manageable forms.

According to the Power Rule for Powers, when you have a term that is raised to another power, you simply multiply the exponents together. For instance, when you see something like \((x^a)^b\), it simplifies to \(x^{a \, \cdot \, b}\).

Applying this rule makes it much easier to work with expressions that involve multiple layers of exponents. In our original exercise, the term \((c^5)^4\) was simplified to \(c^{5 \, \cdot \, 4}\), which equals \(c^{20}\). This step is crucial in breaking down complex algebraic expressions and turning them into simpler forms.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. It makes expressions easier to understand and solve.

In the original exercise, simplification started with distributing the power of 5 to each term inside the parentheses. This means that every part of the expression \((12 c^4 u^3 (w-3)^2)^5\) was raised to the power of 5. This step prepares the expression for further simplification.

Once the powers are distributed, you apply the Power Rule for Powers to make calculations straightforward. For instance, one simplification step took \((w-3)^2\) raised to the power of 5, which became \((w-3)^{2 \, \cdot \, 5}\) or \((w-3)^{10}\).

This transformation allowed us to neatly arrange the expression into individual terms like \(248832c^{20}u^{15}(w-3)^{10}\). The expression was simplified by calculating \(12^5\), which equals 248832, and subsequently combining it with the simplified exponents to create a neat, single expression.

Algebraic Expressions

Algebraic expressions are a way of writing mathematical computations made up of variables, numbers, and operators. They are the foundation of algebraic mathematics.

Algebraic expressions can be simplified using rules like the power rule for powers and principles of arithmetic. In the original problem, working with the expression \(12 c^4 u^3 (w-3)^2\) required manipulation of variables and constants.

Terms in algebraic expressions often consist of variables that may have exponents. These variables stand in for numbers and allow us to solve problems more broadly by inserting any value for these variables later on.

Understanding how to manipulate these expressions involves recognizing patterns and applying mathematical rules effectively. This allows you to transform and simplify expressions as we did in the exercise, ultimately helping us to solve equations or to evaluate expressions easily in different contexts.