Simplifying $\frac{8c^7}{h^3} \times \frac{h^{15}}{16n}$: A Step-by-Step Guide

Introduction to Simplifying Algebraic Expressions

Hey there, math enthusiasts! Let's dive into the fascinating world of algebraic expressions and learn how to simplify one like a pro. Today, we're tackling the expression $\frac{8c^7}{h^3} \times \frac{h^{15}}{16n}$. Don't worry if it looks a bit intimidating at first; we'll break it down step by step, making it super easy to understand. Simplifying algebraic expressions is a fundamental skill in algebra, and it's all about making complex expressions cleaner and more manageable. This not only makes them easier to work with but also helps us to see the underlying relationships between variables and constants more clearly. This process involves applying various algebraic rules and properties, such as the laws of exponents, the commutative and associative properties of multiplication, and the rules for multiplying and dividing fractions. When we simplify an algebraic expression, our goal is to rewrite it in an equivalent form that is as concise as possible. This often involves combining like terms, canceling out common factors, and reducing fractions. The benefits of simplifying are numerous. Firstly, simplified expressions are less prone to errors when performing further calculations. Secondly, they can reveal hidden patterns or relationships that might not be immediately apparent in the original expression. Thirdly, simplifying can make it easier to solve equations and inequalities, as the reduced form often simplifies the steps required to isolate a variable. In our specific expression, $\frac{8c^7}{h^3} \times \frac{h^{15}}{16n}$, we'll focus on simplifying the coefficients, the variables with exponents, and the overall structure of the fraction. This will involve understanding how to multiply fractions, how to handle exponents during multiplication and division, and how to identify and cancel out common factors. This process provides a powerful tool for analyzing and manipulating mathematical relationships, paving the way for advanced mathematical concepts.

Breaking Down the Expression: $\frac{8c^7}{h^3} \times \frac{h^{15}}{16n}$

Now, let's get our hands dirty and break down this expression piece by piece. The expression $\frac{8c^7}{h^3} \times \frac{h^{15}}{16n}$ is a product of two fractions. To simplify it, we will first multiply the numerators (the top parts) and the denominators (the bottom parts) separately. This gives us a new fraction where the numerator is the product of the numerators and the denominator is the product of the denominators. In the numerator, we have $8c^7$ multiplied by $h^{15}$, and in the denominator, we have $h^3$ multiplied by $16n$. So, the first step is to rewrite the expression as a single fraction: $\frac{8c^7 \times h^{15}}{h^3 \times 16n}$. You'll notice that the expression now looks a bit simpler, with all the terms combined into one fraction. Next, we will focus on simplifying the numerator and denominator separately. In the numerator, we can rearrange the terms to group the constants and variables with similar bases. This helps us apply the rules of exponents more easily. In the denominator, we'll do the same to simplify the terms. The goal is to reduce the expression as much as possible by canceling out common factors and combining like terms. This process leverages the fundamental rules of algebra to systematically reduce complex expressions into simpler, more manageable forms. We'll pay close attention to the powers of the variables, such as $h$, and how they interact during multiplication and division. Remember, when multiplying terms with the same base, you add the exponents, and when dividing terms with the same base, you subtract the exponents. This step-by-step approach ensures that we don't miss any simplification opportunities and arrive at the most concise form of the expression.

Step-by-Step Simplification

Multiplying the Numerators and Denominators

Alright, let's start simplifying our expression $\frac{8c^7 \times h^{15}}{h^3 \times 16n}$ step by step. First, focus on the numerator, $8c^7 \times h^{15}$. Since there are no like terms to combine, we simply rewrite it as $8c^7h^{15}$. Next, move to the denominator, $h^3 \times 16n$. Similar to the numerator, we can rewrite this as $16nh^3$. Now, our expression looks like this: $\frac{8c^7h^{15}}{16nh^3}$. The next step involves applying the rules of exponents and simplifying the numerical coefficients. Remember that when multiplying terms with exponents, you add the exponents if the bases are the same, and when dividing, you subtract the exponents. In this case, we have different bases ($c$, $h$, and $n$), so we will need to focus on simplifying the coefficients and the variable $h$, which appears in both the numerator and the denominator. We'll simplify the numerical coefficients by dividing the numerator and the denominator by their greatest common factor (GCF). We will also group the terms with the same base, which will allow us to use the laws of exponents. This involves combining like terms and canceling out common factors to reduce the expression to its simplest form. The systematic application of these rules is crucial to arriving at the final simplified form of the expression, ensuring accuracy and efficiency in our calculations. This process ensures that we correctly apply the rules of algebra, leading to a simplified expression that is mathematically equivalent to the original.

Simplifying Coefficients and Variables

Now, let's focus on simplifying the coefficients and the variable $h$ in the expression $\frac{8c^7h^{15}}{16nh^3}$. First, simplify the coefficients 8 and 16. The greatest common factor of 8 and 16 is 8. Divide both the numerator and denominator by 8: $\frac{8 \div 8}{16 \div 8} = \frac{1}{2}$. This simplifies the expression to $\frac{c^7h^{15}}{2nh^3}$. Next, deal with the variable $h$. We have $h^{15}$ in the numerator and $h^3$ in the denominator. When dividing exponents with the same base, we subtract the exponents: $h^{15} \div h^3 = h^{15-3} = h^{12}$. Replace $h^{15}$ in the numerator with $h^{12}$. Now, the expression becomes $\frac{c^7h^{12}}{2n}$. We have successfully simplified the coefficients and the variable $h$. The remaining variables, $c$ and $n$, have no other terms to combine with, so they remain as they are. This simplified form is equivalent to the original expression, but it's much easier to understand and use in further calculations. This process shows how applying fundamental algebraic rules can transform a complex expression into a simpler, more manageable form. The goal is always to reduce the expression to its most concise form while maintaining its mathematical equivalence.

The Final Simplified Expression

After all the hard work, we've arrived at the final simplified expression: $\frac{c^7h^{12}}{2n}$. Congratulations! We've taken the initial expression, $\frac{8c^7}{h^3} \times \frac{h^{15}}{16n}$, and simplified it into a much cleaner and easier-to-understand form. In this final simplified expression, there are no common factors left to cancel, and all like terms have been combined. The coefficients are simplified, and the exponents of the variables are calculated correctly. The variables $c$, $h$, and $n$ are presented with their respective exponents in the most simplified format. This reduced form is crucial because it allows us to analyze the expression more efficiently. We can now easily see the relationships between the variables and the constant factors. It also means that any further calculations or manipulations will be easier to perform, reducing the risk of making errors. The process of simplifying is a cornerstone of algebra, allowing us to manipulate and understand complex expressions with greater ease and accuracy. Furthermore, understanding simplification is critical for solving equations, working with inequalities, and tackling more complex mathematical problems. This final result is not only correct but also represents the most concise and manageable form of the original expression.

Conclusion

So there you have it! We've successfully simplified the expression $\frac{8c^7}{h^3} \times \frac{h^{15}}{16n}$. This process demonstrates the importance of understanding the rules of exponents, fraction multiplication, and how to combine like terms. By breaking down the problem step by step, we were able to simplify the expression efficiently and accurately. Remember, practice is key when it comes to mastering algebraic simplification. The more you practice, the more comfortable and confident you will become. Keep an eye out for more algebraic expressions to simplify, and keep up the great work. Understanding this process builds a strong foundation for tackling more complex algebraic problems. Mastering these simplification techniques will also help you in other areas of mathematics, like calculus and trigonometry. Keep practicing and exploring different types of algebraic expressions to strengthen your skills. This systematic approach to simplifying algebraic expressions is a valuable skill in mathematics and beyond. It teaches us to break down complex problems into smaller, manageable parts, a skill applicable in many areas of life. So, keep up the practice, and happy simplifying!

For more practice, you can visit the Khan Academy's Algebra Section, they have tons of resources and exercises to help you sharpen your skills.