Problem 96 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(64 y^{18} z^{21}\).

Step 1: Apply the Power to Each Factor

When an entire expression is raised to a power, apply that power to each term inside the parentheses. Hence, we have \((4y^6z^7)^3 = (4)^3 (y^6)^3 (z^7)^3\).

Step 2: Simplify Each Power

Calculate each expression separately: \((4)^3 = 4 \times 4 \times 4 = 64\), \((y^6)^3 = y^{6 \times 3} = y^{18}\), and \((z^7)^3 = z^{7 \times 3} = z^{21}\).

Step 3: Combine the Simplified Expression

Combine all the simplified components from Step 2 into one expression: \(64 y^{18} z^{21}\).

Key Concepts

Power Rule

The power rule is a fundamental rule when dealing with exponents. It comes into play when a power is raised to another power. In this exercise, we have an entire expression, \((4y^6z^7)^3\), raised to the third power. To apply the power rule, distribute the outer power across each factor inside the parentheses. This means the exponent is multiplied with the existing exponents for each factor: for instance, the number 4, the exponent of 6 on \(y\), and 7 on \(z\).

• For the constant 4, the power rule simply cubes it, resulting in \(4^3 = 64\).
• The variable \(y\) with an exponent of 6, becomes \(y^{6 \times 3} = y^{18}\).
• Similarly, \(z\) with an exponent of 7, becomes \(z^{7 \times 3} = z^{21}\).

Understanding the power rule is crucial because it forms the foundation for manipulating expressions involving multiple terms and powers.

Exponents

Exponents represent repeated multiplication, allowing expressions to be more compact and manageable. They are simplified using specific rules, such as the power rule, which we discussed earlier. In the expression \((4y^6z^7)^3\), each part involves exponents:

• The constant 4 is treated with the same rules as variables when raised to a power.
• \(y^6\) and \(z^7\) are variables in terms of exponents, indicating how many times the variable is used as a factor.

In this case, we treat each component within the parentheses separately before recombining them. Through the correct handling of exponents, we can simplify complex expressions into much more manageable forms. Remember that understanding how exponents work aids in simplifying mathematical expressions correctly in various contexts.

Simplification Steps

Simplifying an expression involves reducing it to its simplest form while maintaining its original value. For this problem, the steps are clear:

• **Apply the power to each factor**: Distribute the power to each factor inside the parentheses, turning an expression like \((4y^6z^7)^3\) into \((4)^3\), \((y^6)^3\), and \((z^7)^3\).
• **Simplify each component**: Calculate each term individually: \(4^3 = 64\), \(y^{18}\), and \(z^{21}\). This breaks down computation into manageable pieces and ensures accuracy.
• **Combine simplified results**: Once all parts are simplified, combine them to form the final expression, \(64y^{18}z^{21}\).

Following these steps carefully ensures that each part of the expression is simplified accurately, leading to the correct and final outcome. Simplification is a critical skill in algebra to ensure clarity and correctness in mathematical communication.