Problem 3 - Elementary Algebra

Question

Find each product. $$ y^{6} \cdot y^{4} $$

Step-by-Step Solution

Answer

Answer: The product of \(y^6\) and \(y^4\) is \(y^{10}\).

Step 1: Identify the base and the exponents

The given problem is to find the product of \(y^6\) and \(y^4\). Here, the base is "y" and the exponents are 6 and 4.

Step 2: Use the exponent multiplication rule

With the base "y" and exponents 6 and 4, we will use the exponent multiplication rule such that: $$ y^m * y^n = y^{m+n} $$

Step 3: Plug in the exponents and find the product

Now, we will plug in the exponents (6 and 4) into the exponent multiplication rule: $$ y^6 * y^4 = y^{(6+4)} $$

Step 4: Simplify the expression

Add the exponents: $$ y^{(6+4)} = y^{10} $$ So the product of \(y^6\) and \(y^4\) is: $$ y^6 * y^4 = y^{10} $$

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In our exercise, the algebraic expression includes the variable 'y' raised to different powers: \(y^6\) and \(y^4\).
To fully understand algebraic expressions, it's important to recognize the components involved:

• **Variables**: These are symbols, such as 'y', used to represent numbers that can vary.
• **Constants**: These are numbers on their own without attached variables.
• **Coefficients**: These are numbers multiplied by a variable. For example, in the term \(3x\), the number 3 is the coefficient.
• **Exponents**: Indicate how many times the base (variable) is used as a factor.

Understanding these components helps when manipulating expressions, particularly when dealing with multiplication or simplification of such expressions.

Multiplication of Exponents

Multiplication of exponents occurs when you multiply expressions that have the same base. The underlying rule to remember is simple:

• If you have the same base, you just add the exponents: \(a^m \times a^n = a^{m+n}\).

Let's see this in action with our exercise:

• We're multiplying: \(y^6 \times y^4\).
• Since the base 'y' is the same, we apply the rule: add the exponents (6 + 4).
• The new exponent becomes 10, so the result is \(y^{10}\).

Remember, this rule only applies if the bases are the same. Mixing up different bases requires a different approach, often seen in more advanced exercises.

Simplifying Expressions

After combining like terms, such as multiplying exponents with the same base, you'll often want to simplify the expression further. Simplifying makes expressions easier to understand and work with.
Simplification involves reducing expressions to their simplest form by:

• **Adding or subtracting like terms:** If terms in an expression have the same variables and exponents, combine them.
• **Multiplying exponents with the same base:** As demonstrated in our example, simply add the exponents to simplify.
• **Eliminating complex fractions or factors:** Try to express terms in the simplest fraction possible or remove any unnecessary factors.

By simplifying, you make the expression as compact as possible, aiding in easier interpretation and further algebraic manipulations.