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Square roots are a mathematical operation that find a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 times 3 equals 9.
In algebra, we often encounter square roots in terms of variables as well. The formula we use to express a square root as an exponent is: \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\). This conversion helps simplify expressions, especially when dealing with algebraic operations.
To see this in action, consider the expression \(\sqrt{(x+1)^3}\). Applying the formula, it becomes \((x+1)^{\frac{3}{2}}\).
This technique is crucial as it makes other operations like factoring and simplifying more straightforward, since we're often more familiar with working with exponents than roots.

Factoring expressions is the process of breaking down more complex expressions into simpler parts or "factors" that, when multiplied together, give back the original expression.
When working with algebraic expressions, look for common terms that appear in each part of the expression. This allows you to factor them out, simplifying the entire expression.
In our problem, both terms of the expression \((x+1)^{\frac{3}{2}} + (x+1)^{\frac{5}{2}}\) share the common factor \((x+1)^{\frac{3}{2}}\). Factoring it out leaves:\

• \((x+1)^{\frac{3}{2}}\) which is the common term.
• The simplified terms: \(1 + (x+1)^{1}\).

This step is essential because it reveals a simpler form of the expression that can be manipulated more easily, often paving the way for further simplification or solving equations.

Simplifying algebraic expressions means reducing them to their most condensed form. It's about making the expression as simple as possible to work with, removing any unnecessary complexity.
In the final step of our exercise, the expression inside the parenthesis \(1 + (x+1)^1\) simplifies further by combining like terms:
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• First, expand \((x+1)^1\) to \(x + 1\).
• Then add the numbers: \(1 + x + 1\) which simplifies to \(x + 2\).

Now, multiplying it back with the factored out term \((x+1)^{\frac{3}{2}}\) gives us the fully simplified expression \((x+1)^{\frac{3}{2}}(x+2)\). Simplifying expressions is a fundamental skill in algebra, as it makes further operations tractable and the expression itself easier to understand.