Problem 36 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ (6 y)^{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(216 y^3\).

Step 1: Apply the Power to Each Factor

Start by distributing the exponent to both the constant and the variable inside the parentheses. The expression \((6 y)^3\) implies raising both the numerical part 6 and the variable part \(y\) to the power of 3. Therefore, the expression becomes \(6^3 y^3\).

Step 2: Calculate the Power of the Constant

Calculate \(6^3\). This means multiplying 6 by itself three times: \(6 \times 6 \times 6 = 36 \times 6 = 216\). So, \(6^3 = 216\).

Step 3: Combine the Results

Using the results from the previous steps, combine them into the final expression. You have \(216 y^3\), where 216 is the cube of the constant part and \(y^3\) maintains the variable part.

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they are not that complicated once you understand their meaning. A negative exponent indicates that you need to take the reciprocal of the base. In simpler words, if you see something like \(a^{-n}\), it translates to \( \frac{1}{a^n}\). This concept is crucial because it allows for the simplification of expressions into a form without negative exponents.
Here are some key points to remember about negative exponents:

• Negative exponent means inversion. For example, \(2^{-3}\) becomes \(\frac{1}{2^3} = \frac{1}{8}\).
• Found in expressions, negative exponents appear as part of simplifying or rewriting equations.
• Working with them properly results in a clear, positive exponent expression.

Avoid confusion by rewriting any negative exponents you come across right away. This approach ensures no lingering or mishandled negative terms.

Power of a Product

The power of a product rule makes handling expressions with products under an exponent straightforward. This rule states that when you have a product inside a parenthesis raised to an exponent, you can apply the exponent to each factor in the product separately.
For example, if you have \((ab)^n\), this is the same as \(a^n b^n\). Applying the exponent to each part of the product allows you to separate things nicely and calculate each piece individually.
Consider these parts when applying the power of a product rule:

• Distribute the exponent to each factor within the parentheses.
• Calculate each part individually for straightforward solving.
• Combine all the parts afterward for the final product expression.

This rule simplifies calculations and avoids errors, especially in multi-part expressions. It's a time saver and a clarity booster when simplifying mathematical products.

Polynomials

Polynomials are mathematical expressions involving variables raised to various powers and combined through addition, subtraction, and multiplication. They come in different forms and are an essential part of algebra, appearing frequently in equations, functions, and even calculus.
A polynomial is any expression that looks like \(ax^n + bx^{n-1} + ... + constant\). Here:

• \(a, b, ...\) are coefficients that multiply the variables.
• \(x, y, ...\) represent variables that form the polynomial.
• \(n, n-1,... \) are the exponents on the variables.

Polynomials are crucial because they represent functions and are used to model real-world scenarios. They also set the foundation for solving equations and understanding relationships between quantities.
When simplifying polynomials, you may need to combine like terms—terms that have the same variables raised to the same powers. Understanding how to break down and evaluate polynomials is key to mastering algebra and extending your skills to higher levels of math.