Simplify the expression by first getting a common denominator, then pulling out any common factors in the numerator, and finally expanding and/or combining like terms from the remaining factor.
9(5x + 1)3 sqrtx2 + 1 + (5x + 1)5/ 2 sqrt x2 + 1


Let's simplify the expression step by step.

### Given expression:
\[
9(5x + 1)\frac{3\sqrt{x^2 + 1}}{1} + \frac{(5x + 1)^5}{2\sqrt{x^2 + 1}}
\]

### Step 1: Get a common denominator

We observe that the terms involve different denominators: \(1\) and \(2\sqrt{x^2 + 1}\). The least common denominator (LCD) between them is \(2\sqrt{x^2 + 1}\).

1. For the first term, \(9(5x + 1)\frac{3\sqrt{x^2 + 1}}{1}\), we multiply both the numerator and the denominator by \(2\sqrt{x^2 + 1}\) to match the LCD:
   \[
   9(5x + 1)\frac{3\sqrt{x^2 + 1}}{1} = \frac{9(5x + 1)3\sqrt{x^2 + 1} \cdot 2\sqrt{x^2 + 1}}{2\sqrt{x^2 + 1}} = \frac{18(5x + 1)(x^2 + 1)}{2\sqrt{x^2 + 1}}
   \]

2. The second term, \(\frac{(5x + 1)^5}{2\sqrt{x^2 + 1}}\), already has the same denominator, so we leave it as it is.

### Step 2: Combine the two terms over the common denominator

Now that both terms have the same denominator, we can combine them:
\[
\frac{18(5x + 1)(x^2 + 1) + (5x + 1)^5}{2\sqrt{x^2 + 1}}
\]

### Step 3: Pull out any common factors in the numerator

We can factor out \((5x + 1)\) from both terms in the numerator:
\[
\frac{(5x + 1)\left[18(x^2 + 1) + (5x + 1)^4\right]}{2\sqrt{x^2 + 1}}
\]

### Step 4: Expand and combine like terms

1. Expand \(18(x^2 + 1)\):
   \[
   18(x^2 + 1) = 18x^2 + 18
   \]

2. Keep \((5x + 1)^4\) as it is, as it's a higher power term.

So, the expression becomes:
\[
\frac{(5x + 1)\left[18x^2 + 18 + (5x + 1)^4\right]}{2\sqrt{x^2 + 1}}
\]

This is the simplified form. It involves pulling out the common factor and combining the two terms over a common denominator.

Would you like further details on any of the steps?

Here are some questions to explore further:
1. How do you expand the binomial \((5x + 1)^4\)?
2. What is the significance of factoring out common terms?
3. Can this expression be simplified further by substituting a specific value for \(x\)?
4. How do you deal with square roots in the denominator in general?
5. Could you graph the resulting function to understand its behavior?

**Tip:** When simplifying expressions, always look for opportunities to factor common terms before expanding—it can save time and avoid mistakes.

Simplify the expression by first getting a common denominator, then pulling out any common factors in the numerator, and finally expanding and/or combining like terms from the remaining factor. 9(5x + 1)3 sqrtx2 + 1 + (5x + 1)5/ 2 sqrt x2 + 1

Learn how to simplify complex rational expressions by finding a common denominator, factoring common terms, and expanding polynomials. This step-by-step solution is ideal for high school students in Grades 10-12 studying algebraic expressions and rational functions.

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