Problem 134 - Algebra A Combined Function

Question

Simplify each expression. Assume that variables represent positive integers. $$ \left(3 y^{2 z}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(27 y^{6z}\).

Step 1: Apply the Power Rule

To simplify the expression \(\left(3 y^{2z}\right)^3\), we start by applying the power rule, which states that \((a^m)^n = a^{m \cdot n}\). For each component inside the parentheses, raise it to the power of 3.

Step 2: Simplifying the Coefficient

First, apply the power rule to the coefficient 3, giving us \(3^3\). Calculate \(3^3 = 27\).

Step 3: Simplifying the Exponent Part

Next, apply the power rule to the variable part \(y^{2z}\). When raised to the third power, \(\left(y^{2z}\right)^3\) becomes \(y^{2z \cdot 3}\). Simplify this to \(y^{6z}\).

Step 4: Combine the Results

Combine the results from the previous steps: \(3^3 = 27\) and \(y^{6z}\). Thus, the expression \(\left(3 y^{2z}\right)^3\) simplifies to \(27 y^{6z}\).

Key Concepts

Power Rule

The power rule is a fundamental rule in algebra used for simplifying expressions that involve exponents. It is articulated as

• \( (a^m)^n = a^{m \cdot n} \)

This means if you have a base raised to a power, and that quantity is raised to another power, you can multiply the exponents to simplify it. In the original exercise, we applied the power rule to both the coefficient and the variable part of the expression \( (3 y^{2z})^3 \).

This rule helps streamline the calculations by reducing complex exponential expressions into simpler forms. By applying this rule, mathematical operations become more manageable, especially when dealing with multiple layers of exponents.

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. They are essential in algebra because they allow for more concise expression of operations involving multiplication. An exponent is written as a small number to the right and above the base number. For example, in \(3^3\), 3 is the base and 3 is the exponent, meaning \(3 \times 3 \times 3\).

In our simplification task, we had \(3^3\) which simplifies to 27. The exponent indicates how many times the base number is used as a factor in multiplication. Another example involves the expression \(y^{2z}\), where 2z is the exponent. This means that \(y\) is multiplied by itself \(2z\) times. Understanding how to work with exponents is crucial because they frequently appear in algebraic expressions and equations.

Variables

Variables in algebra are symbols that represent unknown or arbitrary numbers and are fundamental in forming algebraic expressions. Typically, letters such as x, y, and z are used to denote variables. In the expression provided, 'y' and 'z' are variables.

Variables allow us to generalize mathematical statements and solve problems where the numerical values are not directly known. They are strategic placeholders that can take on varying values, facilitating problem-solving and algebraic manipulations. In the original exercise, the variable 'y' is raised to the exponent \(2z\), which itself involves a variable 'z'. This implies that the expression can change shape depending on the values substituted for these variables. By understanding how variables function within expressions, you gain the flexibility to tackle a wide range of mathematical problems.