Problem 44 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(-3 x^{7} y z^{2}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(-27x^{21} y^3 z^6\).

Step 1: Identify the Expression

The given expression is \( \left(-3 x^{7} y z^{2}\right)^{3} \). It involves a power applied to a product of terms. We'll need to apply the power rule and power of a product rule to simplify it.

Step 2: Apply the Power Rule to Each Factor

The power rule states that \( (a^m)^n = a^{m imes n} \). Apply this rule to each factor inside the parentheses: - \((-3)^3\)- \((x^7)^3\)- \((y)^3\)- \((z^2)^3\)

Step 3: Calculate \((-3)^3\)

Calculate the cube of -3: \[(-3)^3 = (-3) \times (-3) \times (-3) = -27.\]

Step 4: Exponentiate \(x^7\) using the Power Rule

Using the power rule, raise \(x^7\) to the 3rd power: \[(x^7)^3 = x^{7 \times 3} = x^{21}\]

Step 5: Exponentiate \(y\) using the Power Rule

Raise \(y\) to the 3rd power: \[(y)^3 = y^3\]

Step 6: Exponentiate \(z^2\) using the Power Rule

Raise \(z^2\) to the 3rd power: \[(z^2)^3 = z^{2 \times 3} = z^6\]

Step 7: Combine All Simplified Components

Combine all the simplified components together: \[-27 \cdot x^{21} \cdot y^3 \cdot z^6\]Thus, the simplified expression is: \[-27x^{21} y^3 z^6\]

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. For example, the expression \(x^7\) means \(x\) multiplied by itself 7 times. Exponents are fundamental in algebra as they provide a cleaner way to write large repeated multiplication.
They make it easier to work with derivatives, integrals, and various algebraic manipulations. The power rule for exponents helps in handling expressions with powers efficiently.
By using the power rule, you can break down complicated expressions into simple components. This simplification process is essential in calculus and polynomial solutions, allowing easier computation and understanding of algebraic relationships.

Exploring Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. In our exercise, we have an expression \((-3 x^7 y z^2)^3\).
Each part of this expression plays a role:

• \(-3\) is a constant multiplier.
• \(x^7\) is a term with a variable raised to a power.
• \(y\) and \(z^2\) are also variable terms, each having their own exponents or lack thereof.

When dealing with algebraic expressions, it is crucial to understand how each component interacts with the others.
You should know how to apply the rules of exponents to simplify and manipulate these expressions efficiently. This understanding lays the groundwork for solving even more complex algebraic problems.

The Power of a Product Rule

The Power of a Product Rule is a handy tool in algebra. It states that when raising a product to a power, you can apply the exponent to each factor within the product individually.
Mathematically, it's expressed as \((ab)^n = a^n b^n\). In our example \((-3 x^7 y z^2)^3\),
you apply the power of 3 to each individual part:

• \((-3)^3\)
• \((x^7)^3\)
• \((y)^3\)
• \((z^2)^3\)

This allows the expression to be broken down into smaller, manageable sections. You then calculate each term separately for easier processing. Understanding this rule simplifies the handling of complicated terms, which is especially beneficial in algebraic simplifications and solving equations.