Simplifying Algebraic Expressions A Step By Step Guide To $5 Z^2(10-8 Z^4)$

In mathematics, simplifying expressions is a fundamental skill that allows us to rewrite complex equations into more manageable forms. This often involves applying the distributive property, combining like terms, and using the order of operations. In this article, we will delve into the process of simplifying the expression $5z^2(10-8z^4)$. This involves distributing the term outside the parenthesis to each term inside the parenthesis and then combining any like terms. Understanding these steps is crucial for solving more complex algebraic problems and is a key concept in algebra. Let's break down each step to make the simplification process clear and straightforward.

Understanding the Distributive Property

At the heart of simplifying the expression $5z^2(10-8z^4)$ lies the distributive property. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms inside a set of parentheses. In simpler terms, it states that for any numbers a, b, and c:

a(b + c) = ab + ac

This property is essential because it enables us to remove parentheses from expressions, which is often the first step in simplifying them. In our case, we have $5z^2$ multiplied by the binomial $(10 - 8z^4)$. To apply the distributive property, we need to multiply $5z^2$ by each term inside the parentheses individually. This means we will multiply $5z^2$ by 10 and then $5z^2$ by $-8z^4$. Understanding and correctly applying the distributive property is crucial for simplifying algebraic expressions and solving equations. It's a building block for more advanced algebraic concepts and is used extensively in various mathematical fields. By mastering this property, you can confidently tackle more complex problems and gain a deeper understanding of algebraic manipulations. The distributive property not only simplifies expressions but also helps in solving equations and understanding the structure of mathematical relationships.

Applying the Distributive Property to the Expression

Now, let's apply the distributive property to our expression, $5z^2(10-8z^4)$. As discussed, we need to multiply $5z^2$ by each term inside the parentheses. First, we multiply $5z^2$ by 10:

$5z^2 * 10 = 50z^2$

Next, we multiply $5z^2$ by $-8z^4$. Remember to pay attention to the signs and the rules of exponents. When multiplying terms with the same base, we add their exponents:

$5z^2 * -8z^4 = -40z^(2+4) = -40z^6$

So, after applying the distributive property, our expression becomes:

$50z^2 - 40z^6$

This step is critical because it eliminates the parentheses, allowing us to further simplify the expression if possible. By carefully applying the distributive property, we have successfully expanded the original expression into a sum of two terms. This transformation is a key step in simplifying and solving algebraic problems. The ability to correctly distribute is essential for manipulating equations and finding solutions. It is a foundational skill that is used repeatedly in algebra and calculus. Understanding the nuances of the distributive property, such as handling negative signs and exponents, is crucial for mastering algebraic simplification.

Combining Like Terms and Final Simplification

After applying the distributive property, our expression looks like this: $50z^2 - 40z^6$. The next step in simplifying is to check for like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have two terms: $50z^2$ and $-40z^6$. Notice that the variable $z$ is raised to different powers in each term (2 and 6, respectively). This means that $50z^2$ and $-40z^6$ are not like terms.

Since there are no like terms to combine, the expression is already in its simplest form. However, it is conventional to write polynomials in descending order of exponents. This means we should rearrange the terms so that the term with the highest exponent comes first.

Therefore, the simplified expression is:

$-40z^6 + 50z^2$

This is the final simplified form of the original expression. By rearranging the terms, we have presented the expression in a standard format that is easy to read and understand. The process of checking for like terms is crucial in simplification because it ensures that the expression is written in its most concise form. Recognizing and combining like terms is a skill that is frequently used in algebra and is essential for solving equations and simplifying more complex expressions. The final step of arranging the terms in descending order of exponents is a matter of convention and helps to maintain consistency in mathematical notation.

Final Answer

In conclusion, to simplify the expression $5z^2(10-8z^4)$, we followed these steps:

1. Applied the distributive property to multiply $5z^2$ by each term inside the parentheses.
2. Simplified each term by multiplying the coefficients and adding the exponents of like variables.
3. Checked for like terms and found that there were none to combine.
4. Rearranged the terms in descending order of exponents for clarity.

Thus, the simplified answer is:

$-40z^6 + 50z^2$

This process demonstrates how algebraic expressions can be simplified using fundamental properties and operations. Mastering these techniques is essential for success in algebra and higher-level mathematics. Simplification not only makes expressions easier to understand but also facilitates further calculations and problem-solving. The ability to simplify expressions efficiently is a valuable skill that can save time and reduce errors in mathematical work. Understanding the underlying principles and practicing these techniques will build confidence and proficiency in algebra. The steps outlined in this article provide a clear and systematic approach to simplifying algebraic expressions, ensuring that each step is performed accurately and efficiently. By following these steps, you can confidently simplify a wide range of expressions and improve your algebraic skills.