Problem 94 - Intermediate Algebra

Question

Simplify each exponential expression. $$ \left(4 y^{6} z^{7}\right)^{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(64y^{18}z^{21}\).

Step 1: Apply the Power of a Power Rule

The expression \( (4y^{6}z^{7})^{3} \) involves raising a power to another power. Apply the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\). This rule applies individually to \(4\), \(y^6\), and \(z^7\).

Step 2: Simplify the Coefficient

Raise the coefficient 4 to the power of 3. This calculation is straightforward: \(4^3 = 64\).

Step 3: Simplify the Exponents of the Variables

For \(y^6\), raise the exponent by multiplying: \((y^6)^3 = y^{18}\). Similarly, for \(z^7\), calculate \((z^7)^3 = z^{21}\).

Step 4: Combine the Simplified Parts

Put together all simplified parts that resulted from the above steps. The simplified expression is \(64 y^{18} z^{21}\).

Key Concepts

Power of a Power Rule

When dealing with exponential expressions, the power of a power rule is a fundamental concept. This rule is relevant when you have an exponent raised to another exponent, such as \((a^m)^n\). It states that you can multiply the exponents together: \((a^m)^n = a^{m \cdot n}\). This is very helpful in simplifying complex expressions more effectively.

In practical terms, think of the power of a power rule as distributing the outer exponent to each part of the term inside the parenthesis. For example, in the expression \((4y^6z^7)^3\), each part gets affected by the power of 3:

• The base 4 is raised to the third power: \(4^3\).
• The exponent of \(y\) inside is multiplied by 3: \((y^6)^3 = y^{6 \cdot 3}\).
• Similarly for \(z\), \((z^7)^3 = z^{7 \cdot 3}\).

By understanding and applying this rule, complex exponential expressions can become simpler and less intimidating to work with.

Simplifying Exponents

Simplifying exponents means reducing expressions with exponents to their simplest form. This process involves applying rules of exponents systematically, such as the power of a power rule, as discussed earlier.

For example, consider \((y^6)^3\). By multiplying the exponents, we simplify it to \(y^{18}\). Similarly, \((z^7)^3\) becomes \(z^{21}\). Treating the expression \(4^3\) as a separate base with an exponent, we compute \(4^3 = 64\).

Simplifying each part like this can turn complicated exponential expressions into more manageable ones. It's like breaking a big task into smaller, doable pieces. Remember these steps:

• Calculate the powers of any numerical coefficients.
• Multiply any exponents from the inside with exterior powers.
• Combine all parts into the simplest form.

Proper simplification makes future calculations easier and clearer.

Algebraic Expressions

Understanding algebraic expressions is essential for anyone studying mathematics, especially with exponents involved. These expressions include variables, constants, and mathematical operations.

Expressions like \(4y^6z^7\) include a constant coefficient (4) and variables with exponents (\(y^6\) and \(z^7\)). When such expressions are raised to a power, simplifying them demands a comprehensive understanding of how to manipulate coefficients and exponents using established rules.

Working with algebraic expressions involves:

• Understanding coefficients and how they scale with exponents.
• Recognizing patterns and applying rules, like the power of a power rule, effectively.
• Combining simplified elements strategically to form the simplest expression.

By mastering these basic skills, you can simplify complex algebraic expressions efficiently, making them easier to analyze and solve.