Simplify Expression With Positive Exponents: (16x^2y^7)/(40x^8y^3)

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Hey guys! Today, we're diving into simplifying algebraic expressions, specifically focusing on how to handle exponents like pros. We'll break down the steps to simplify the expression (16x2y7) / (40x8y3), ensuring we only use positive exponents in our final answer. This is a common type of problem you'll see in algebra, so mastering it is super important. Let's get started!

Understanding the Basics of Exponents

Before we jump into the main problem, let's quickly review the fundamental rules of exponents. These rules are the backbone of simplifying any expression with exponents, so make sure you're comfortable with them. We'll be using these rules extensively throughout this article.

  1. Product of Powers: When multiplying terms with the same base, you add the exponents. Mathematically, this is expressed as a^m * a^n = a^(m+n). For example, if you have x^2 * x^3, you add the exponents 2 and 3 to get x^(2+3) = x^5. This rule is crucial when you're combining like terms in an expression.

  2. Quotient of Powers: When dividing terms with the same base, you subtract the exponents. This rule is written as a^m / a^n = a^(m-n). For instance, if you have x^5 / x^2, you subtract the exponents 2 from 5 to get x^(5-2) = x^3. This rule is especially helpful when simplifying fractions with exponents, like the one we're tackling today.

  3. Power of a Power: When raising a power to another power, you multiply the exponents. This rule is represented as (am)n = a^(mn). For example, if you have (x2)3, you multiply the exponents 2 and 3 to get x^(23) = x^6. This rule is essential when dealing with expressions enclosed in parentheses with exponents outside.

  4. Power of a Product: When raising a product to a power, you distribute the exponent to each factor in the product. This rule is written as (ab)^n = a^n * b^n. For example, if you have (2x)^3, you distribute the exponent 3 to both 2 and x to get 2^3 * x^3 = 8x^3. This rule is useful when simplifying expressions where multiple terms are raised to a power.

  5. Power of a Quotient: When raising a quotient to a power, you distribute the exponent to both the numerator and the denominator. This rule is expressed as (a/b)^n = a^n / b^n. For example, if you have (x/y)^2, you distribute the exponent 2 to both x and y to get x^2 / y^2. This rule is important when dealing with fractions raised to a power.

  6. Zero Exponent: Any non-zero number raised to the power of 0 is equal to 1. This is written as a^0 = 1 (where a ≠ 0). For example, x^0 = 1 and 5^0 = 1. This rule might seem simple, but it's essential for simplifying expressions where variables or numbers are raised to the power of zero.

  7. Negative Exponents: A term raised to a negative exponent is equal to its reciprocal with a positive exponent. This rule is written as a^(-n) = 1/a^n. For example, x^(-2) = 1/x^2. Dealing with negative exponents is a key part of simplifying expressions, as we often need to rewrite them as positive exponents.

These exponent rules are the tools we'll use to simplify our expression. Got them down? Great! Let's move on to tackling our problem.

Breaking Down the Expression: (16x2y7) / (40x8y3)

Okay, let's dive into the expression we need to simplify: (16x2y7) / (40x8y3). The first thing we want to do is break this down into smaller, more manageable parts. Think of it like separating the coefficients (the numbers) from the variables (x and y). This will make the simplification process much clearer.

We can rewrite the expression as:

(16/40) * (x2/x8) * (y7/y3)

See how we've separated the numerical part (16/40) and the variable parts (x2/x8 and y7/y3)? This is a crucial step in simplifying complex expressions. By isolating each component, we can apply the exponent rules more effectively.

Now, let's look at each part individually. First, we'll simplify the numerical part, then we'll tackle the x terms, and finally, the y terms. By addressing each component separately, we minimize the chances of making mistakes and keep the process organized.

Simplifying the Coefficients: 16/40

The first part of our expression to tackle is the coefficient fraction: 16/40. Simplifying fractions is all about finding the greatest common divisor (GCD) of the numerator and the denominator, and then dividing both by that GCD. This will reduce the fraction to its simplest form.

In this case, both 16 and 40 are divisible by 8. So, the greatest common divisor (GCD) of 16 and 40 is 8. Now, we divide both the numerator and the denominator by 8:

16 ÷ 8 = 2

40 ÷ 8 = 5

Therefore, the simplified fraction is 2/5. This means that the numerical part of our expression is now neatly simplified. Remember, keeping the numbers in their simplest form from the beginning makes the rest of the simplification process much easier and less prone to errors.

So, we've taken the first step in simplifying our expression by reducing the coefficients. Now, let's move on to the variable parts and apply those exponent rules we talked about earlier.

Simplifying the x Terms: x^2 / x^8

Next up, we have the x terms: x^2 / x^8. This is where the quotient of powers rule comes into play. Remember, the quotient of powers rule states that when dividing terms with the same base, you subtract the exponents: a^m / a^n = a^(m-n).

Applying this rule to our x terms, we subtract the exponent in the denominator (8) from the exponent in the numerator (2):

x^(2-8) = x^(-6)

So, we have x^(-6). But wait! Our goal is to express the final answer using only positive exponents. What do we do with that negative exponent? This is where the rule for negative exponents comes in handy.

Recall that a term raised to a negative exponent is equal to its reciprocal with a positive exponent: a^(-n) = 1/a^n. Applying this to x^(-6), we get:

x^(-6) = 1/x^6

Therefore, the simplified form of the x terms, using only positive exponents, is 1/x^6. It's crucial to remember this step of converting negative exponents to positive ones to fully simplify the expression.

Now that we've simplified the x terms, let's move on to the y terms and see how the quotient of powers rule applies there as well.

Simplifying the y Terms: y^7 / y^3

Now let's tackle the y terms: y^7 / y^3. Just like with the x terms, we'll use the quotient of powers rule here. Again, this rule tells us that when dividing terms with the same base, we subtract the exponents: a^m / a^n = a^(m-n).

Applying the rule to our y terms, we subtract the exponent in the denominator (3) from the exponent in the numerator (7):

y^(7-3) = y^4

So, we get y^4. Notice that this time, the result has a positive exponent, so we don't need to do any further manipulation. This is exactly what we want! It means we can directly include y^4 in our final simplified expression.

Simplifying the y terms was pretty straightforward, right? Now, we've simplified all the individual components of our original expression: the coefficients, the x terms, and the y terms. The next step is to put it all back together.

Putting It All Together: The Final Simplified Expression

We've done the hard work of simplifying each part of the expression individually. Now comes the satisfying part: putting it all back together to get our final answer! Let's recap what we've found:

  • The simplified coefficient is 2/5.
  • The simplified x terms are 1/x^6.
  • The simplified y terms are y^4.

Now, we just need to multiply these simplified parts together:

(2/5) * (1/x^6) * y^4

To combine these, we multiply the numerators together and the denominators together:

(2 * 1 * y^4) / (5 * x^6)

This simplifies to:

2y^4 / 5x^6

And there you have it! Our final simplified expression, using only positive exponents, is 2y^4 / 5x^6. This is the fully simplified form of the original expression (16x2y7) / (40x8y3).

Key Takeaways and Practice

Wow, we covered a lot in this guide! The key to simplifying expressions with exponents is to break them down into smaller parts, apply the exponent rules systematically, and always remember to express your final answer with positive exponents.

Let's recap the main steps we followed:

  1. Break Down the Expression: Separate the coefficients and the variables.
  2. Simplify Coefficients: Find the GCD and reduce the fraction.
  3. Apply Quotient of Powers Rule: Subtract exponents when dividing terms with the same base.
  4. Handle Negative Exponents: Convert them to positive exponents using the reciprocal rule.
  5. Combine Simplified Terms: Multiply the simplified coefficients and variables together.

To really master these skills, practice is essential. Try simplifying similar expressions on your own. You can change the coefficients, exponents, or even add more variables to make the problems more challenging. The more you practice, the more confident you'll become in simplifying expressions with exponents.

Simplifying expressions with exponents might seem tricky at first, but with a solid understanding of the rules and a bit of practice, you'll be simplifying like a pro in no time! Keep up the great work, and don't hesitate to revisit this guide whenever you need a refresher. You got this!