Problem 33 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 5 x^{2} y^{0}+3 x^{2} y+2 x^{2} y+1, \quad y \neq 0 \quad \text { (Look closely at the exponents.) } $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$5x^2 \cdot x^0 + 3x^2y + 2x^2y + x^0$$ Answer: The simplified expression is $$5x^2 + 5x^2y + 1$$.

Step 1: Rewrite the expression with the zero exponent

In the given expression, we have a variable with an exponent of 0. According to the rules of exponents, any non-zero number raised to the power of 0 is equal to 1. Therefore, we can rewrite the expression as: $$ 5 x^{2} \cdot 1+3 x^{2} y+2 x^{2} y+1 $$

Step 2: Combine the like terms

Now, we need to combine the like terms, which are terms with the same variables raised to the same exponent. In this expression, we can combine the terms with \(x^2 y\): $$ 5 x^{2}+ (3 x^{2} y+2 x^{2} y)+1 $$

Step 3: Simplify the combined terms

When we add the terms with \(x^2 y\), we get the following simplified expression: $$ 5 x^{2}+ 5 x^2 y + 1 $$ So the final simplified expression is: $$ 5 x^{2}+ 5 x^2 y + 1 $$

Key Concepts

Understanding the Rules of Exponents

The rules of exponents are essential for simplifying algebraic expressions. When dealing with exponents, there are a few key rules to remember:

• Any non-zero base raised to the power of zero equals 1. This rule helps simplify expressions like \( y^0 \), as seen in the original exercise, which becomes 1.


• For multiplication with the same base, simply add the exponents, e.g., \( x^a \cdot x^b = x^{a+b} \).


• In division, subtract the exponent in the denominator from the exponent in the numerator, such as \( \frac{x^a}{x^b} = x^{a-b} \), provided \( x eq 0 \).

Applying these rules, the expression \( 5x^2y^0 \) simplifies to \( 5x^2 \cdot 1 \) because \( y^0 = 1 \). Knowing and using these rules will make simplifying expressions more straightforward and efficient.

Combining Like Terms in Algebraic Expressions

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms have the same variable part, meaning the same variables raised to identical powers.

• In the expression \( 3x^2y + 2x^2y \), both terms are like as they share the variable part \( x^2y \).


• To combine them, add or subtract their coefficients, which are the numerical parts. Here, it becomes \( (3 + 2)x^2y = 5x^2y \).


• Remember, only coefficients and not the variables or their exponents change during this combination.

This method helps in organizing expressions into fewer terms, making them more manageable and easier to interpret. Always ensure the terms are exactly alike before combining; otherwise, they need to stay separate.

Simplification Steps for Expressions

Simplification involves several steps to make expressions easier to work with.

• Start by applying the rules of exponents to handle any zero exponents or powers present.


• Next, systematically combine like terms. Identify terms with the same variable parts and add their coefficients.


• The expression from the original exercise went from \( 5x^2y^0 + 3x^2y + 2x^2y + 1 \) to \( 5x^2 \cdot 1 + 5x^2y + 1 \).

These steps lead to a simplified form \( 5x^2 + 5x^2y + 1 \), which is concise and straightforward. Simplification not only makes expressions easier to understand but also prepares them for further manipulations in algebra.