Problem 89 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(-6 x y z^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(36x^2y^2z^6\).

Step 1: Apply the Power Rule

The expression we are given is \((-6xyz^3)^2\). Applying the power rule \((a^m)^n = a^{m\cdot n}\), we must raise each factor inside the parentheses to the power of 2: \((-6)^2\), \(x^2\), \(y^2\), and \((z^3)^2\).

Step 2: Simplify Each Factor

Calculate each power individually. First, \((-6)^2 = 36\). Next, \(x^2 = x^2\) and \(y^2 = y^2\). Finally, use the power rule again for the \(z\) term: \((z^3)^2 = z^{3\cdot 2} = z^6\).

Step 3: Combine Simplified Factors

Combine all the simplified parts: \(36\) for \((-6)^2\), \(x^2\), \(y^2\), and \(z^6\). The fully simplified expression is \(36x^2y^2z^6\).

Key Concepts

Power Rule

The power rule is a fundamental principle in algebra that simplifies the process of working with exponents. It states that when you have an exponent raised to another exponent, you multiply the two exponents together. For example, \[ (a^m)^n = a^{m \cdot n} \]This is exactly what we did with the original expression \[(-6xyz^3)^2\].Each part of the expression inside the parentheses has been raised to the power of two.

• For numbers: We calculate \((-6)^2\), which equals 36.
• For variables with their own exponents, we multiply their exponent by 2. So for \(z^3\), it becomes \(z^{3 \cdot 2} = z^6\).

This rule makes it easier to handle more complex expressions that require exponentiation.

Exponentiation

Exponentiation is the process of raising a number or expression to a power. Essentially, it indicates how many times you multiply a number by itself. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\). This is similar to what happens in the expression \((-6)^2\).We apply the exponent to both the coefficient (-6) and each of the variables,

• For the coefficient: Raising \((-6)^2\) results in 36.
• For each variable: The process of exponentiation is represented as \(x^2, y^2, z^6\) in the simplified expression.

By understanding rules of exponentiation, it becomes easier to simplify complex algebraic expressions like our original example.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They represent mathematical phrases or sentences that can include variables (like \(x, y, z\)), coefficients, and operators (+, -, \/, \*).In the problem we tackled: \[ (-6xyz^3)^2 \]

• The algebraic expression contains the coefficient (-6) and variables \(x, y, z\).
• The power applied to the entire expression requires us to simplify each part, applying properties such as the power rule along the way.

Simplifying algebraic expressions often involves applying rules and properties systematically, as shown in the step-by-step solution.

Mathematical Simplification

Mathematical simplification involves rewriting an expression in its most straightforward or comprehensible form. The goal is to reduce the expression without changing its value. Using simplification strategies, you can transform complex expressions into simpler terms.

• In the case of \((-6xyz^3)^2\), simplification took these steps: applying the power rule followed by exponentiation.
• Once simplified, the expression reads \(36x^2y^2z^6\), which is compact and easier to understand or work with.

Through simplification, the task becomes less daunting, making it easier to solve problems or perform further operations on the expression.