Problem 14 - Elementary Algebra

Question

\(\left(5 y^{7}\right)\left(6 y^{3}\right)\)

Step-by-Step Solution

Answer

30y^{10}

Step 1: Multiply the Coefficients

The coefficients are the numerical parts of the terms. Here, they are 5 and 6. Multiply these coefficients: 5 * 6 = 30.

Step 2: Apply the Power Rule for Exponents

When multiplying terms with the same base, add their exponents. For the variable part, add the exponents of y: 7 + 3 = 10.

Step 3: Combine the Results

Combine the product of the coefficients from Step 1 and the result of the exponent addition from Step 2 to form the final expression: 30y^{10}.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase combining numbers, variables, and operators (like addition or multiplication). For example, in the expression \(5y^7\), 5 is the coefficient (numerical part), and \(y^7\) is the variable part. Similarly, \(6y^3\) has 6 as the coefficient and \(y^3\) as the variable part. Algebraic expressions allow us to describe mathematical situations in a general way. They can be simplified and manipulated using various algebraic rules, making complex calculations easier and more manageable.

Exponents and Powers

Exponents and powers are crucial components in algebra. An exponent tells us how many times to multiply a base number by itself. In the term \(y^7\), y is the base, and 7 is the exponent. So, \(y^7 = y \times y \times y \times y \times y \times y \times y\). Similarly, \(y^3\) means \(y \times y \times y\). When we multiply expressions with the same base, we have a specific rule:

• Power Rule: When multiplying terms with the same base, add their exponents.

For instance, when we multiply \(y^7\) and \(y^3\), we add the exponents: \(y^{7+3} = y^{10}\). This rule simplifies the multiplication of terms and helps in performing calculations more efficiently.

Multiplication of Terms

Multiplying algebraic terms involves combining both the coefficients and the variable parts. Let's break it down using the given exercise:

• Step 1: Multiply the coefficients, the numerical parts. Here, 5 and 6. So, \(5 \times 6 = 30\)
• Step 2: Use the power rule to add the exponents of the variable part y. For \(y^7\) and \(y^3\), add the exponents: \(7 + 3 = 10\).
• Step 3: Combine the results. The final expression is \(30y^{10}\).

This simple three-step approach makes it easy to multiply algebraic terms. Start with the coefficients, then handle the exponents, and finally, combine everything for the result.