Problem 30 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ \left(3 y^{2}\right)\left(4 y^{5}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Multiply the Coefficients

We start by multiplying the numerical coefficients of each term. The coefficient of the first term is 3 and the coefficient of the second term is 4. Therefore, we multiply 3 and 4 together: \[3 \times 4 = 12\]

Step 2: Apply the Law of Exponents

When multiplying expressions with the same base, we add the exponents according to the exponent rules. Here, the base is \(y\) with exponents 2 and 5, respectively. \[y^{2} \times y^{5} = y^{2+5} = y^{7}\]

Step 3: Combine the Results

Combine the results from Steps 1 and 2 to write the simplified expression. Since the coefficient is 12 and the new power of \(y\) is 7, the expression becomes:\[12y^{7}\]

Key Concepts

Coefficient

In algebra, the term **coefficient** refers to the number multiplied by a variable in an expression. It tells you how many times you have the base variable. For example, in the expression \(3y^2\), 3 is the coefficient, and it indicates that the variable \(y^2\) is present three times.

• The coefficient can be a positive or negative number, an integer, a fraction, or even a decimal.


• It is always found right in front of the variable it is multiplying.

In our expression \((3y^2)\cdot(4y^5)\), the coefficients are 3 and 4. So, multiplying them together, as shown in Step 1, gives us a new coefficient of 12. This shows how coefficients combine when multiplying expressions.

Exponent Rules

Exponents are essential in algebra for expressing repeated multiplication. **Exponent rules** help us to simplify expressions efficiently.
When dealing with exponents, there are a few key rules to remember:

• Product of Powers Rule: When multiplying like bases, you add the exponents. For example, \(x^a \times x^b = x^{a+b}\).


• Quotient of Powers Rule: When dividing like bases, subtract the exponents. \(x^a \div x^b = x^{a-b}\).


• Power of a Power Rule: When raising a power to a power, multiply the exponents. \((x^a)^b = x^{a\times b}\).

In our given problem \((3y^2)\cdot(4y^5)\), we used the Product of Powers Rule. Since both expressions share the base \(y\), we add the exponents: 2 and 5, resulting in the simplified expression \(y^7\). Following these rules makes working with exponents straightforward.

Simplifying Expressions

**Simplifying expressions** involves reducing them to their most basic form. This often means combining like terms and using arithmetic operations to make calculations easier to interpret and solve.

• It involves performing operations like addition, subtraction, multiplication, and following the rules of exponents.


• By simplifying, the expression becomes clearer and more manageable.

In the original exercise \((3y^2)\cdot(4y^5)\), simplifying the expression involves three main steps:
1. **Multiply the coefficients**: Combine the numbers 3 and 4, which gives 12.
2. **Apply the exponent rule for multiplying like bases**: Since the base \(y\) is common, add exponents 2 and 5 to get \(y^7\).
3. **Combine these results**: Yielding the fully simplified expression \(12y^7\).
By engaging in these steps, we clarify the expression and make it easier to handle further algebraic operations.