Problem 41 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ 3 y^{2} \cdot(2 y)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(6 y^{5}\)

Step 1: Simplify the Coefficients

Multiply the coefficients 3 and 2 together to get 6. This leaves the expression as \(6 y^{2} \cdot y^{3}\)

Step 2: Apply the Property of Exponents

When multiplying exponents with the same base, you add the powers. In this case, we add the exponents of 2 and 3 to get 5. The final expression is \(6 y^{5}\)

Key Concepts

Understanding Powers

A power is a way to express repeated multiplication of the same number by itself. The expression is written in the form of a base raised to an exponent. Here, the base is the number being multiplied, and the exponent tells you how many times to multiply the base by itself. For example, in the expression \(y^2\), "y" is the base, and "2" is the exponent, indicating that "y" is multiplied by itself one more time: \(y \times y\).
Understanding powers is essential as they are used to simplify expressions and make multiplication of the same number more manageable. Powers are not only limited to whole numbers; they can also be applied to algebraic terms, such as in our example with variables and coefficients.

Applying the Product of Powers

When multiplying powers with the same base, a specific rule called the Product of Powers Rule is used. This rule states that you simply add the exponents while keeping the base unchanged. This is because you are essentially stacking the repeated multiplications together. For instance, when you multiply \(y^2\) by \(y^3\), you are essentially adding up the number of times "y" is multiplied, resulting in \(y^{2+3}\) or \(y^5\).
This process makes it easier to work with expressions involving powers and helps in shortening complex multiplications into more manageable forms. By mastering this rule, you'll be able to simplify expressions efficiently.

Simplifying Expressions with Powers

Simplifying expressions involves making them easier to work with or understand by following mathematical laws and rules. When dealing with powers, as in the expression \(3 y^{2} \cdot (2 y)^{3}\), you first simplify any constants. Here, you multiply the coefficients (3 and 2) to obtain 6. Once the coefficients are combined, pay attention to the variable base with its exponents. Using the Product of Powers Rule, add up the exponents (2 and 3 in this case) to get 5, leading to the simplified form \(6y^5\).
Simplifying expressions through power rules lets you operate with complex problems more easily. A streamlined expression is much easier to analyze and understand, especially when variables and coefficients are interacting in a problem.