Problem 34 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \left(3 y^{2}\right)\left(4 y^{5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(12y^7\).

Step 1: Expand the Multiplication

Write the expression as a multiplication of coefficients and variables separately: \[ (3 \cdot 4) \times (y^2 \cdot y^5) \]

Step 2: Multiply the Coefficients

Multiply the numeric coefficients together: \[ 3 \times 4 = 12 \]

Step 3: Apply the Product of Powers Rule

Use the product of powers rule, which states \( a^m \times a^n = a^{m+n} \), to combine the variable powers:\[ y^2 \times y^5 = y^{2+5} = y^7 \]

Step 4: Combine Results

Combine the results of the coefficients and the variable parts:\[ 12y^7 \]

Key Concepts

Product of Powers Rule

When you encounter problems involving multiplying terms with the same base, like variables, the Product of Powers Rule is your go-to tool. This rule states that when you multiply two exponents with the same base, you simply add the exponents together. This can be expressed as:

• If you have terms like \(a^m \times a^n\), you'll get \(a^{m+n}\).

In our exercise, you have \(y^2\) and \(y^5\). Both parts contain the base \(y\). Using the Product of Powers Rule, we add the exponents, resulting in \(y^{2+5} = y^7\). It's like adding up all the little power boosts \(y\) gets in the multiplication. When tackling similar problems, always check that the bases are the same before applying this rule.

Simplifying Expressions

Simplifying expressions means breaking complex mathematical expressions into simpler, reduced forms. This often involves combining like terms and using mathematical rules effectively. In our exercise, simplifying revolved around the expression \((3y^2)\times(4y^5)\).
First, we expressed multiplication of like terms using separate factors: \((3 \cdot 4) \times (y^2 \cdot y^5)\). This separates coefficients from the mathematical variables, which helps clarify what we are working with.
After simplifying, the expression is easier to handle or compute in future steps. By clearly understanding each part of the expression, you can reduce errors and quickly find the simplest form during tests or homework.

Multiplying Coefficients

Multiplying coefficients is just like multiplying regular numbers. In mathematics, coefficients are the numerical parts in terms that are usually multiplied with variables or other values. In our case, 3 and 4 are the coefficients associated with the variable part of the expression.

• Start by multiplying the coefficients separately: \(3 \cdot 4 = 12\).
• This step isolates the numbers, allowing you to handle the variables separately.

By multiplying the coefficients first, you simplify that part of the expression and focus next on handling the variables. This clear separation helps both in managing complex problems and avoiding mistakes during calculations.