Problem 76 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Write your answer as a power. $$ \left(8 x^{2} y^{8}\right)^{3} $$

Step-by-Step Solution

Answer

After simplification, \((8x^2y^8)^3\) simplifies to \(512x^6y^{24}\).

Step 1: Apply Exponent to the Numeric Part

First, raise 8 to the power of 3. This gives \(8^3 = 512\). This is based on basic exponent rules where the exponent is applied to the numeric base.

Step 2: Apply Exponent to the Variables

Next, raise each of the variables to the power of 3 individually. Using the power of a power rule, for \(x^2\) raised to the power of 3 will give \(x^{2*3} = x^6\). For \(y^8\) raised to the power of 3 will give \(y^{8*3} = y^{24}\).

Step 3: Combine the Results

Finally, combine the results of the previous steps in multiplication form. After computation from steps 1 and 2, the results were 512, \(x^6\) and \(y^{24}\). Combining these gives \(512x^6y^{24}\).

Key Concepts

Power of a Power Rule

The power of a power rule is a handy tool in exponentiation. It helps simplify expressions when you have an exponent applied to another exponent. To use this rule, multiply the exponents together. For example, when simplifying \((x^2)^3\), you multiply the exponents (2 and 3) to get \(x^{6}\). This process makes mathematical expressions easier to handle, especially when the expressions become complex.

• Helps reduce the complexity of expressions with multiple exponents.
• Uses multiplication between exponents for simplification.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They enable us to represent mathematical relationships in a general form. For instance, when you see \(8x^2y^8\), this is an algebraic expression with variables \(x\) and \(y\), and a constant \(8\).

• Made up of numbers and variables.
• Can represent more complex relationships using simple terms.

Understanding algebraic expressions is essential. They provide the foundation for solving equations and inequalities in mathematics.

Basic Exponent Rules

Basic exponent rules form the backbone of working with powers. These rules help simplify expressions and solve equations effectively. One key rule is applying exponents to numbers and variables separately, as shown in \((8x^2y^8)^3\). Each component — the constant and the variables — are raised to the power individually.

• When multiplying powers with the same base, add the exponents.
• When raising a power to another power, multiply the exponents.
• The zero exponent rule states that any non-zero number to the power of zero is 1.

By applying these rules, expressions can be drastically simplified, making complex calculations more manageable.