Problem 40 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(6 x^{4}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified form of \((6x^4)^2\) is \(36x^8\).

Step 1: Distribute the Exponent

Apply the power rule which states that \((a^{m})^{n} = a^{m*n}\) to the expression. Here \(a=6\), \(m=1\) and \(n=2\). Also, \(a=x\), \(m=4\) and \(n=2\). When you distribute the square to each element inside the curtain, you get: \( (6^{1 * 2}) * (x^{4 * 2}) \)

Step 2: Simplify the Expression

By simplifying the expression you get: \(36x^8\)

Key Concepts

Power Rule

The power rule is a fundamental concept when working with exponential expressions. This rule simplifies the process of raising a power to another power. According to this rule, when you have an expression in the form \[(a^m)^n\]it is equivalent to \[a^{m imes n}\]This means that you multiply the exponents together. Apply the power rule to each component of an expression individually. For example, in the expression \[(6x^4)^2\]apply the power rule as follows:

• For the constant 6, where the implied exponent is 1, you calculate: \(6^{1 \times 2} = 6^2\).
• For the variable \(x\), you calculate: \(x^{4 \times 2} = x^8\).

Simplifying Expressions

Simplifying expressions is key in making them more manageable and easier to work with. It involves reducing expressions to their simplest form without changing their value. This process often includes:

• Combining like terms
• Applying algebraic rules such as the distributive property
• Reducing coefficients and exponents
• Eliminating unnecessary parentheses

For exponential expressions, simplification usually involves using the power rule to adjust exponents and then calculating any remaining operations. In the given exercise, after applying the power rule to \[(6x^4)^2\]we use basic arithmetic to further simplify, yielding \[36x^8\]This includes calculating \(6^2\) to get 36, and recognizing that \(x^8\) reflects the double-exponentiation processed earlier, making the expression concise and neat.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In algebra, a major focus is on learning how to manipulate these expressions using mathematical rules to solve equations or simplify terms. They can represent real-world problems or abstract mathematical ideas. Components of algebraic expressions include:

• Constants, like the number 6 in our exercise
• Variables, symbolized by letters such as \(x\)
• Coefficients, which are numbers that multiply variables
• Exponents, which indicate how many times the base is used in a multiplication
• Operations, such as addition, subtraction, multiplication, and division

Understanding how to use and simplify algebraic expressions is essential for solving equations. In the given exercise, \[(6x^4)^2\]is an algebraic expression that we simplify using the power rule, leading to the more manageable form \[36x^8\]This highlights the process of taking complex expressions and making them easier to interpret and use in further computations.