Equivalent Expressions: Simplify $10x^9$ And $(60x^{-6})^{-1}$

In this article, we will explore how to simplify expressions involving exponents and multiplication. Specifically, we'll tackle the problem of finding expressions equivalent to the product of $10x^9$ and $(60x^{-6})^{-1}$, assuming $x eq 0$. This involves understanding the rules of exponents, negative exponents, and how to handle coefficients. Let's dive in and break down the steps to solve this problem.

Understanding the Problem

Before we jump into solving, let's make sure we understand what the question is asking. We're given two expressions: $10x^9$ and $(60x^{-6})^{-1}$. We need to find their product and then identify which of the given options are equivalent to this product. The condition $x eq 0$ is crucial because it prevents us from dividing by zero, which is undefined in mathematics. The key here is simplifying the second expression and then multiplying it by the first expression.

Breaking Down the Expressions

Let's start by simplifying the second expression, $(60x^{-6})^{-1}$. Remember that a negative exponent means we take the reciprocal of the base raised to the positive exponent. In other words, a^{-n} = rac{1}{a^n}. Also, when we raise a product to a power, we raise each factor in the product to that power. So, we have:

$(60x^{-6})^{-1} = 60^{-1} (x^{-6})^{-1}$

Now, we apply the rule of negative exponents and the power of a power rule, which states that $(a^m)^n = a^{mn}$. We get:

60^{-1} (x^{-6})^{-1} = rac{1}{60} x^{(-6)(-1)} = rac{1}{60} x^6

So, the second expression simplifies to rac{1}{60} x^6. Now, we need to multiply this by the first expression, $10x^9$.

Calculating the Product

Now that we've simplified $(60x^{-6})^{-1}$ to rac{1}{60} x^6, we can find the product of $10x^9$ and rac{1}{60} x^6. When multiplying expressions with the same base, we add the exponents. The coefficients are multiplied directly. So, we have:

10x^9 imes rac{1}{60} x^6 = 10 imes rac{1}{60} imes x^9 imes x^6

First, multiply the coefficients:

10 imes rac{1}{60} = rac{10}{60} = rac{1}{6}

Next, multiply the variables with the same base by adding their exponents:

$x^9 imes x^6 = x^{9+6} = x^{15}$

Combining these results, we get:

rac{1}{6} x^{15}

This is the simplified product of the two given expressions. Now, let's compare this result with the provided options.

Comparing with the Options

We have simplified the product to rac{1}{6} x^{15}. Now, let's evaluate the given options to see which ones are equivalent:

A. rac{10 x^9}{60 x^{-6}} B. rac{10 x^9}{60 x^6} C. rac{x^{15}}{6} D. $6 x^{15}$

Let's analyze each option:

Option A: rac{10 x^9}{60 x^{-6}}

To simplify this expression, we divide the coefficients and subtract the exponents. Remember that dividing by a term with a negative exponent is the same as multiplying by the term with a positive exponent. So, we have:

rac{10 x^9}{60 x^{-6}} = rac{10}{60} imes rac{x^9}{x^{-6}} = rac{1}{6} imes x^{9 - (-6)} = rac{1}{6} x^{9+6} = rac{1}{6} x^{15}

This is the same as our simplified product, so option A is equivalent.

Option B: rac{10 x^9}{60 x^6}

Simplifying this expression, we get:

rac{10 x^9}{60 x^6} = rac{10}{60} imes rac{x^9}{x^6} = rac{1}{6} imes x^{9-6} = rac{1}{6} x^3

This is not the same as rac{1}{6} x^{15}, so option B is not equivalent.

Option C: rac{x^{15}}{6}

This can be rewritten as:

rac{x^{15}}{6} = rac{1}{6} x^{15}

This is the same as our simplified product, so option C is equivalent.

Option D: $6 x^{15}$

This is not the same as rac{1}{6} x^{15}, so option D is not equivalent.

Conclusion

After simplifying the product of $10x^9$ and $(60x^{-6})^{-1}$, we found that the equivalent expressions are:

• Option A: rac{10 x^9}{60 x^{-6}}
• Option C: rac{x^{15}}{6}

Therefore, the correct answers are A and C. Understanding the rules of exponents, including negative exponents and the power of a power rule, is crucial for simplifying algebraic expressions. By breaking down the problem step by step, we were able to find the equivalent expressions efficiently. Remember to always simplify expressions before comparing them to options to avoid errors. Mastering these concepts will greatly help in tackling more complex mathematical problems.

For further learning and practice on exponents and algebraic expressions, you can visit Khan Academy's Algebra I section. They offer comprehensive lessons and practice exercises to reinforce your understanding.