Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression. a. \(\left(2 x^{3} y^{2}\right)^{3}\) b. \(\left(4 x^{2} y^{2} z^{3}\right)^{2}\)

Step-by-Step Solution

Answer

a. \(8x^9y^6\) b. \(16x^4y^4z^6\)

Step 1: Apply rule 1

Raise each factor to the power of 3: \( (2^3) (x^3)^3 (y^2)^3 \)

Step 2: Apply rule 2

Simplify the exponents: \( 2^3 x^{3 \times 3} y^{2 \times 3} \)

Step 3: Calculate the final result

Compute the exponents: \( 8x^9y^6 \) The simplified expression for part a is \( 8x^9y^6 \). b. \(\left(4 x^{2} y^{2} z^{3}\right)^{2}\)

Step 1: Apply rule 1

Raise each factor to the power of 2: \( (4^2) (x^2)^2 (y^2)^2 (z^3)^2 \)

Step 2: Apply rule 2

Simplify the exponents: \( 4^2 x^{2 \times 2} y^{2 \times 2} z^{3 \times 2} \)

Step 3: Calculate the final result

Compute the exponents: \( 16x^4y^4z^6 \) The simplified expression for part b is \( 16x^4y^4z^6 \).

Key Concepts

Exponentiation Rules

Understanding exponentiation and how to apply its rules is essential when simplifying expressions. Exponentiation involves raising a number or expression to a power, which reflects how many times you multiply the base by itself. When simplifying expressions, the following exponentiation rules are particularly helpful:

• Power of a Product Rule: For any integers \(a\), \(m\), and \(n\), \((a \, b)^m = a^m \, b^m\). This means that when you raise a product to a power, you raise each factor of the product to that power.
• Power of a Power Rule: For any integers \(a\) and \(m, n\), \((a^m)^n = a^{m\times n}\). To apply this, multiply the exponents when you have a power raised to another power.

Keeping these rules in mind allows you to handle complex expressions with confidence, simplifying them into more manageable forms.

Algebraic Manipulation

Algebraic manipulation is the skill of rearranging and simplifying expressions using algebraic rules. Here’s how you can effectively carry out this task:

• Identify and apply the correct exponentiation rules. For instance, if an expression includes a product raised to a power, use the power of a product rule.
• Simplify step by step. Breaking complex expressions into simpler parts can help ensure accuracy and clarity in your computations.
• Perform calculations systematically. Carry out the multiplication or division of terms raised to powers separately, which can lessen errors.

Proper algebraic manipulation allows you to progressively break down a problem into understandable parts, which makes solving algebraic problems much easier.

Power of a Power Property

The "power of a power" property is a powerful shortcut when dealing with nested exponents. By mastering this rule, you can simplify expressions quickly and efficiently.
Here’s how it works: if you have a term like \((a^m)^n\), the power of a power property tells you to multiply the exponents, transforming it into \(a^{mn}\). This rule simplifies the expression by directly allowing you to calculate the new exponent.

• If the base is a constant, compute its power first before handling the variables with exponents, as seen in the expression \(2^3 x^{3 \times 3} y^{2 \times 3}\).
• Apply this property to every component in an expression separately. This makes calculating the final simplified form straightforward.
• Check your work by ensuring the resulting exponent makes sense in the context of your calculations.

By using the power of a power property, you streamline the simplification process, making complex expressions much less daunting.