Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blank. A. To simplify \(\left(2 y^{3} z^{2}\right)^{4},\) what factors within the parentheses must be raised to the fourth power? B. To simplify \(\left(\frac{y^{3}}{z^{2}}\right)^{4},\) what two expressions must be raised to the fourth power?

Step-by-Step Solution

Answer

For A: 2, \(y^3\), \(z^2\). For B: \(y^3\), \(z^2\).

Step 1: Understanding the Problem Part A

We need to simplify \( (2y^3z^2)^4 \). This means each component inside the parentheses, i.e., the numerical coefficient and each variable's power, must be raised to the fourth power.

Step 2: Applying the Exponent Rule for Part A

According to the power of a product rule, \((ab)^n = a^n b^n\). Apply this to each part in \((2y^3z^2)^4\). The coefficient 2 and each of the variables \(y^3\) and \(z^2\) should be raised to the fourth power.

Step 3: Simplification for Part A

Raise each component to the fourth power: \(2^4, (y^3)^4, (z^2)^4\). This results in \(16, y^{12}, z^{8}\). Hence, the simplified expression is \(16y^{12}z^{8}\).

Step 4: Understanding the Problem Part B

Here, we need to simplify \( \left( \frac{y^3}{z^2} \right)^4 \). This involves applying the exponent to both the numerator and the denominator individually.

Step 5: Applying the Exponent Rule for Part B

According to the quotient of powers rule, \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). Apply this to \( \frac{y^3}{z^2} \), raising both \(y^3\) and \(z^2\) to the fourth power.

Step 6: Simplification for Part B

Raise the numerator and denominator to the fourth power: \((y^3)^4\) becomes \(y^{12}\), and \((z^2)^4\) becomes \(z^8\). Therefore, the simplified expression is \(\frac{y^{12}}{z^8}\).

Key Concepts

Power of a Product Rule

The Power of a Product Rule is a fundamental concept in exponentiation. It simplifies expressions where a product is raised to a power. When we have an expression like \[ (ab)^n \] we apply the rule by raising each element inside the parentheses to the power of \( n \). In mathematical terms, this results in \[ a^n \, b^n \].

Let's look at an example to clarify: Suppose we want to simplify \( (2y^3z^2)^4 \). Here, we apply the Power of a Product Rule to each part inside the parentheses to get:

• \( 2^4 \) for the numerical coefficient
• \( (y^3)^4 \) for the variable \( y \)
• \( (z^2)^4 \) for the variable \( z \)

These result in \( 16y^{12}z^8 \) as the final simplified expression. This rule allows us to manage complex expressions easily by tackling each component individually and then combining the results.

Quotient of Powers Rule

The Quotient of Powers Rule deals with dividing powers that share the same base. It enables us to simplify expressions where a fraction is raised to a power. When we have an expression like \[ \left( \frac{a}{b} \right)^n \], we apply the rule by raising both the numerator and the denominator to the power of \( n \). Mathematically, this becomes:\[ \frac{a^n}{b^n} \].

For example, if we consider \( \left( \frac{y^3}{z^2} \right)^4 \), we use the Quotient of Powers Rule as follows:

• Raise the numerator \((y^3)^4\) resulting in \(y^{12}\)
• Raise the denominator \((z^2)^4\) resulting in \(z^8\)

So, the simplified expression is \( \frac{y^{12}}{z^8} \). This rule breaks down complex fractions into more manageable parts, thereby simplifying calculations significantly.

Simplification of Algebraic Expressions

Simplification of algebraic expressions is a critical skill in mathematics that allows us to write expressions in their most concise form. The key is to apply rules of exponents effectively to reduce the complexity of expressions.

When faced with expressions like \( (2y^3z^2)^4 \) and \( \left( \frac{y^3}{z^2} \right)^4 \), simplification involves applying the relevant exponent rules, such as the Power of a Product Rule and the Quotient of Powers Rule, respectively. By doing this:

• We avoid long and cumbersome calculations.
• The expression becomes easier to work with in further algebraic manipulations.

Understanding these steps helps in simplifying more complex algebraic expressions accurately. It is about taking each component, raising it to the appropriate power, and combining the results effectively. Mastering simplification is key to success in algebra and beyond.