Simplify Complex Expression: Step-by-Step Guide

by Alex Johnson 48 views

Have you ever encountered a math expression that looks like it belongs more in a spaceship control panel than a textbook? Don't worry; you're not alone! Complex expressions can seem intimidating, but with a systematic approach, they can be tamed. This guide will walk you through the process of simplifying a particularly interesting expression, breaking it down into manageable steps. So, grab your calculator (or just your brain!), and let's dive in!

The Expression at Hand

Before we start, let's take a look at the mathematical puzzle we're going to solve:

4−(cos⁡a−2)2a2+2a+1÷1−1a2+1a\frac{4-(\cos a -2)^2}{a^2+2 a+1} \div \frac{1-\frac{1}{a^2+1}}{a}

It looks a bit scary, doesn't it? But don't fret! We'll break it down piece by piece, applying fundamental mathematical principles along the way. The key to simplifying any complex expression lies in recognizing patterns, applying algebraic identities, and performing operations in the correct order. We'll tackle the numerator, denominator, and division separately, and then bring it all together for the grand finale – the simplified expression!

Step 1: Simplify the Numerator

Let's begin by focusing on the numerator of the first fraction: 4−(cos⁡a−2)24-(\cos a -2)^2. This part of the expression involves a squared term, which hints at a potential algebraic identity we can use. Specifically, we can treat this as a difference of squares pattern, but first, let's expand the squared term:

(cos⁡a−2)2=(cos⁡a−2)(cos⁡a−2)=cos⁡2a−4cos⁡a+4(\cos a - 2)^2 = (\cos a - 2)(\cos a - 2) = \cos^2 a - 4\cos a + 4

Now, substitute this back into the numerator:

4−(cos⁡2a−4cos⁡a+4)=4−cos⁡2a+4cos⁡a−44 - (\cos^2 a - 4\cos a + 4) = 4 - \cos^2 a + 4\cos a - 4

Notice that the 4 and -4 cancel each other out, leaving us with:

−cos⁡2a+4cos⁡a-\cos^2 a + 4\cos a

We can factor out a cos⁡a\cos a from this expression:

cos⁡a(4−cos⁡a)\cos a (4 - \cos a)

This is a crucial step in simplifying the numerator. By expanding the square and then factoring, we've transformed a seemingly complex term into a more manageable form. This kind of algebraic manipulation is a cornerstone of simplifying mathematical expressions. Identifying patterns and applying appropriate identities can drastically reduce the complexity of the problem. Remember, practice makes perfect! The more you work with these kinds of expressions, the quicker you'll be able to spot the opportunities for simplification. Keep an eye out for differences of squares, perfect square trinomials, and other common algebraic patterns. Also, don't be afraid to break the problem down into smaller parts, as we've done here. Focusing on one section of the expression at a time can make the overall task seem much less daunting. In the next step, we'll move on to simplifying the denominator, so stay tuned!

Step 2: Simplify the Denominator of the First Fraction

Now, let's turn our attention to the denominator of the first fraction: a2+2a+1a^2 + 2a + 1. This quadratic expression looks suspiciously like a perfect square trinomial. Do you remember what those are? A perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form is x2+2xy+y2=(x+y)2x^2 + 2xy + y^2 = (x + y)^2 or x2−2xy+y2=(x−y)2x^2 - 2xy + y^2 = (x - y)^2.

In our case, we have a2+2a+1a^2 + 2a + 1. We can see that the first term is a2a^2, the last term is 11 (which is 121^2), and the middle term is 2a2a (which is 2∗a∗12 * a * 1). This perfectly fits the pattern of a perfect square trinomial!

Therefore, we can factor the denominator as follows:

a2+2a+1=(a+1)2a^2 + 2a + 1 = (a + 1)^2

This is another vital step in simplifying the expression. Recognizing perfect square trinomials is a key skill in algebra, and it can significantly simplify complex expressions. By factoring the denominator, we've made it much easier to work with. This will be especially helpful when we combine the fractions later on. Remember to always be on the lookout for these patterns! Perfect square trinomials are common in mathematical expressions, and being able to quickly identify and factor them will save you a lot of time and effort. The ability to recognize patterns like this comes with practice, so keep working on your algebra skills! In the next step, we'll tackle the second fraction and continue our journey towards simplifying the entire expression. We're making great progress, so keep up the good work!

Step 3: Simplify the Second Fraction

Next, let's focus on the second fraction: 1−1a2+1a\frac{1-\frac{1}{a^2+1}}{a}. This fraction has a fraction within a fraction, which can look a bit messy. The key to simplifying this is to get rid of the inner fraction. To do this, we need to find a common denominator for the terms in the numerator.

The numerator is 1−1a2+11 - \frac{1}{a^2 + 1}. We can rewrite 1 as a2+1a2+1\frac{a^2 + 1}{a^2 + 1} to have a common denominator:

a2+1a2+1−1a2+1\frac{a^2 + 1}{a^2 + 1} - \frac{1}{a^2 + 1}

Now that we have a common denominator, we can subtract the fractions:

a2+1−1a2+1=a2a2+1\frac{a^2 + 1 - 1}{a^2 + 1} = \frac{a^2}{a^2 + 1}

So, the numerator of the second fraction simplifies to a2a2+1\frac{a^2}{a^2 + 1}. Now we can rewrite the entire second fraction:

a2a2+1a\frac{\frac{a^2}{a^2 + 1}}{a}

To divide by a fraction, we multiply by its reciprocal. In this case, we're dividing by aa, which can be written as a1\frac{a}{1}. So, we multiply by 1a\frac{1}{a}:

a2a2+1⋅1a=a2a(a2+1)\frac{a^2}{a^2 + 1} \cdot \frac{1}{a} = \frac{a^2}{a(a^2 + 1)}

We can simplify this further by canceling out a factor of aa from the numerator and denominator:

aa2+1\frac{a}{a^2 + 1}

This simplification is crucial because it transforms the complex fraction into a much more manageable form. By finding a common denominator, subtracting the fractions, and then multiplying by the reciprocal, we've eliminated the nested fraction and made the expression easier to work with. This technique is essential for dealing with complex fractions, and mastering it will greatly improve your ability to simplify mathematical expressions. The process of simplifying complex fractions often involves several steps, but by taking it one step at a time, you can break down the problem and arrive at the solution. Remember, practice is key! The more you work with these kinds of fractions, the more comfortable you'll become with the process. In the next step, we'll combine the simplified fractions and see if we can simplify the expression even further. We're almost there, so keep going!

Step 4: Combine the Simplified Fractions

Now that we've simplified both the first and second fractions, we can combine them. Remember the original expression:

4−(cos⁡a−2)2a2+2a+1÷1−1a2+1a\frac{4-(\cos a -2)^2}{a^2+2 a+1} \div \frac{1-\frac{1}{a^2+1}}{a}

We've simplified the numerator of the first fraction to cos⁡a(4−cos⁡a)\cos a (4 - \cos a), the denominator of the first fraction to (a+1)2(a + 1)^2, and the second fraction to aa2+1\frac{a}{a^2 + 1}. Let's substitute these simplified expressions back into the original expression:

cos⁡a(4−cos⁡a)(a+1)2÷aa2+1\frac{\cos a (4 - \cos a)}{(a + 1)^2} \div \frac{a}{a^2 + 1}

To divide fractions, we multiply by the reciprocal of the second fraction:

cos⁡a(4−cos⁡a)(a+1)2⋅a2+1a\frac{\cos a (4 - \cos a)}{(a + 1)^2} \cdot \frac{a^2 + 1}{a}

Now we can multiply the numerators and the denominators:

cos⁡a(4−cos⁡a)(a2+1)a(a+1)2\frac{\cos a (4 - \cos a)(a^2 + 1)}{a(a + 1)^2}

At this point, it's tempting to try to expand the numerator and denominator, but let's take a step back and see if there are any common factors we can cancel out first. Sometimes, expanding everything can make the expression more complicated, not less.

In this case, there aren't any obvious factors that cancel out. The term (a2+1)(a^2 + 1) in the numerator doesn't have any corresponding factors in the denominator. Similarly, the term (4−cos⁡a)(4 - \cos a) doesn't have any matching factors. So, we've reached the simplest form of the expression.

This step highlights the importance of combining the simplified parts correctly. After all the hard work of simplifying each fraction individually, it's crucial to put them together accurately. Remember that dividing by a fraction is the same as multiplying by its reciprocal, and be careful when multiplying the numerators and denominators. Once you've combined the fractions, take a moment to look for common factors that can be canceled out. This can often lead to further simplification. However, in some cases, like this one, there may not be any further simplification possible. Recognizing when you've reached the simplest form is an important skill in itself. In the final step, we'll present the simplified expression and reflect on the entire process. We're almost at the finish line, so let's wrap things up!

Step 5: The Simplified Expression

After all the hard work, we've finally arrived at the simplified expression:

cos⁡a(4−cos⁡a)(a2+1)a(a+1)2\frac{\cos a (4 - \cos a)(a^2 + 1)}{a(a + 1)^2}

This is the most simplified form of the original expression. We've successfully navigated through the complex fractions, factored expressions, and applied algebraic identities to arrive at this result.

Congratulations! You've successfully simplified a complex mathematical expression. This process demonstrates the power of breaking down a problem into smaller, manageable steps. By focusing on each part of the expression individually, we were able to apply the appropriate techniques and simplify it step by step.

Key Takeaways:

  • Break it down: Complex expressions can be intimidating, but by breaking them down into smaller parts, you can make the problem much more manageable.
  • Look for patterns: Recognizing patterns like perfect square trinomials and differences of squares can greatly simplify the expression.
  • Simplify step by step: Simplify each part of the expression before combining them. This will help you avoid errors and make the process easier.
  • Check for common factors: After combining the fractions, look for common factors that can be canceled out.

Simplifying mathematical expressions is a fundamental skill in mathematics. It requires a combination of algebraic knowledge, pattern recognition, and careful execution. By mastering these techniques, you'll be well-equipped to tackle even the most challenging mathematical problems.

So, the next time you encounter a complex expression, remember this step-by-step guide. With a little patience and practice, you'll be able to simplify it like a pro! And remember, math is not just about finding the right answer; it's about the journey of discovery and the satisfaction of solving a challenging problem. Keep practicing, keep exploring, and keep simplifying!

For more information on simplifying mathematical expressions, you can visit Khan Academy's Algebra section.