Problem 40 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ y^{4} \cdot \frac{3 x^{2}}{y^{2}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(3x^{2}y^{2}\).

Step 1: Identify the given expression

The given expression is: $$ y^{4} \cdot \frac{3 x^{2}}{y^{2}} $$

Step 2: Multiply the coefficients and variables separately

When multiplying an expression like this, we multiply the coefficients together and multiply each variable raised to their respective powers together. In this case: - The coefficients are 1 (since the coefficient of \(y^4\) is not explicitly written, it is understood to be 1) and 3. - The variables are \(y^4\) and \(x^2\). So, we have: $$ (1 \cdot 3)(y^{4} \cdot x^{2}) \cdot \frac{1}{y^{2}} $$

Step 3: Simplify the coefficients and group the variables together

We first simplify the product of the coefficients 1 and 3. Next, we'll write the result as a single fraction with a common denominator by multiplying the variables with their respective powers. $$ 3 \cdot \frac{y^{4} x^{2}}{y^{2}} $$

Step 4: Apply the rules of exponents for division

Now, we will simplify the expression using the rule of exponents for division: \(a^{m} / a^{n} = a^{(m-n)}\). In this case, we have \(y^4 / y^2\), so we subtract the exponents, resulting in \(y^{(4-2)}\) or \(y^2\). Our expression now becomes: $$ 3x^{2}y^{2} $$ The final simplified form of the given expression is: $$ 3x^{2}y^{2} $$

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For instance, in the number \( y^4 \), the base \( y \) is multiplied by itself 4 times: \( y \cdot y \cdot y \cdot y \). Exponential notation is a powerful tool in algebra as it simplifies expressions and calculations. When working with exponents, there are several key rules to remember:

• Multiplying with the same base: Add the exponents. For example, \( a^m \cdot a^n = a^{m+n} \).
• Dividing with the same base: Subtract the exponents. This means \( a^m / a^n = a^{m-n} \).
• Power of a power: Multiply the exponents, such as \( (a^m)^n = a^{m\cdot n} \).

In our exercise, these rules help us simplify the expression by dealing with the same base \( y \) in both the numerator and the denominator. Particularly, the rule of division helps change \( y^4/y^2 \) into \( y^{4-2} \), resulting in \( y^2 \). Just remember these rules as they make working with exponents much easier.

Multiplication

Multiplication in algebra is fairly straightforward but can involve multiple components when dealing with algebraic expressions. In expressions like the one in the exercise \( y^4 \cdot \frac{3x^2}{y^2} \), you need to multiply both numbers (coefficients) and variables separately:

• Coefficients: These are the numerical parts of a term. In our exercise, you multiply the implicit 1 from \( y^4 \) by the 3 in \( \frac{3x^2}{y^2} \), resulting in 3.
• Variables: Multiply variables with the same base by adding their exponents. Here, there are no \( x \)'s in \( y^4 \), so the outcome involves placing \( x^2 \) alongside the rest of the expression untouched during this step.

The multiplication steps in algebra often serve to combine like terms and simplify expressions for further calculus or algebraic manipulations. Always ensure you deal with coefficients first and variables next for clarity.

Division

Division in algebra, especially when dealing with exponents, involves simplifying expressions by reducing terms. In the exercise \( y^4 \cdot \frac{3x^2}{y^2} \), division comes into play in the expression \( \frac{y^4}{y^2} \). The rule used here is that when dividing terms with the same base, you subtract the exponents:

• \( a^m / a^n = a^{m-n} \)

Applying this rule simplifies the terms by effectively reducing the power of \( y \) in the quotient. So, \( y^4 / y^2 \) becomes \( y^{4-2} \), which simplifies to \( y^2 \).
This not only reduces the complexity of the expression but also eliminates unnecessary terms.
Understanding the division rules and simplifying where possible is crucial in reducing expressions to their simplest form, making equations easier to handle, solve, or graph. Whenever you handle such divisions, just remember that subtraction of exponents is your go-to tool.