Equivalent Expressions Simplifying 10a^10 / 2a^2

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It's like learning the basic grammar of a language – it allows us to understand and manipulate equations effectively. In this article, we will dissect the algebraic expression $\frac{10 a^{10}}{2 a^2}$ and identify the equivalent expression from the given options. We will delve into the properties of exponents and division to arrive at the correct solution, ensuring a clear and comprehensive understanding of the underlying principles.

Let's start by analyzing the given expression: $\frac{10 a^{10}}{2 a^2}$. This expression involves both numerical coefficients and variables with exponents. Our goal is to simplify this expression into a more concise and manageable form. We can approach this by addressing the numerical coefficients and the variable terms separately.

First, consider the numerical coefficients: 10 and 2. We can simplify the fraction $\frac{10}{2}$ by performing the division operation. 10 divided by 2 equals 5. So, we have simplified the numerical part of the expression to 5.

Next, we turn our attention to the variable terms: $a^{10}$ and $a^2$. These terms involve the variable 'a' raised to different powers. To simplify these terms, we need to recall the quotient rule of exponents. This rule states that when dividing exponential expressions with the same base, we subtract the exponents. Mathematically, this can be expressed as:

$\frac{a^m}{a^n} = a^{m-n}$

Applying this rule to our expression, we have:

$\frac{a^{10}}{a^2} = a^{10-2} = a^8$

Therefore, the variable part of the expression simplifies to $a^8$.

Now, combining the simplified numerical coefficient and the simplified variable term, we get the simplified expression:

$5a^8$

This is the simplified form of the original expression. Now, let's examine the given options and see which one matches our simplified expression.

We are presented with four options, each representing a different expression. Our task is to determine which of these options is equivalent to the simplified expression we derived, which is $5a^8$. Let's examine each option in detail:

• Option A: $8 a \frac{10}{2}$

  This option contains a numerical coefficient of 8 and the variable 'a' multiplied by the fraction $\frac{10}{2}$. Simplifying the fraction, we get 5. So, the expression becomes 8a * 5, which simplifies to 40a. This expression is not equivalent to $5a^8$, as the coefficient and the exponent of 'a' are different. Therefore, Option A is incorrect.

• Option B: $5 a \frac{10}{2}$

  This option has a numerical coefficient of 5 and the variable 'a' multiplied by the fraction $\frac{10}{2}$. Again, simplifying the fraction, we get 5. So, the expression becomes 5a * 5, which simplifies to 25a. This expression is also not equivalent to $5a^8$, as the exponent of 'a' is different. Therefore, Option B is incorrect.

• Option C: $8 a^{10-2}$

  This option has a numerical coefficient of 8 and the variable 'a' raised to the power of (10-2). Simplifying the exponent, we get 10-2 = 8. So, the expression becomes $8a^8$. While the exponent of 'a' matches our simplified expression, the numerical coefficient is different. Therefore, Option C is incorrect.

• Option D: $5 a^{10-2}$

  This option has a numerical coefficient of 5 and the variable 'a' raised to the power of (10-2). Simplifying the exponent, we get 10-2 = 8. So, the expression becomes $5a^8$. This expression perfectly matches our simplified expression. Therefore, Option D is the correct answer.

After careful analysis of each option, we have determined that Option D: $5 a^{10-2}$ is the correct answer. This expression is equivalent to the given expression $\frac{10 a^{10}}{2 a^2}$. We arrived at this conclusion by applying the quotient rule of exponents and simplifying the numerical coefficients and variable terms.

To solve this problem, we utilized the following key concepts:

• Quotient Rule of Exponents: This rule states that when dividing exponential expressions with the same base, we subtract the exponents.

• Simplifying Fractions: We simplified the fraction formed by the numerical coefficients.

• Combining Like Terms: We combined the simplified numerical coefficient and the simplified variable term to obtain the final expression.

Simplifying algebraic expressions is a crucial skill in mathematics. By understanding the properties of exponents and applying them correctly, we can effectively manipulate and simplify complex expressions. In this article, we successfully simplified the expression $\frac{10 a^{10}}{2 a^2}$ and identified the equivalent expression from the given options. This exercise highlights the importance of understanding fundamental mathematical principles and applying them systematically to solve problems. Mastering these skills will undoubtedly enhance your mathematical proficiency and problem-solving abilities.