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When we talk about writing a mathematical expression in "expanded form," we're basically breaking it down into a longer, detailed version. Imagine you're unraveling a math problem so you can see all its parts clearly. In the given exercise, we have this sigma notation: \[ \sum_{k=1}^{4} \sqrt{k} \] This might look a bit complex at first, but it simply means, "add up the square roots of all numbers from 1 to 4." By expressing it in expanded form, we're transforming it from a compact expression into a more understandable form. Here's how it works:

• Identify each value of \( k \) - in this case, values from 1 to 4.
• Calculate the square root of each \( k \).
• List these individual calculations separately, showing exactly what you're summing.

The expanded form of our example is \( \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} \). Breaking it down this way helps you see each part of the sum, making it easier to work through or verify one's work.

A step-by-step solution is like a guided tour through a math problem. It leads you through each action so you understand not just what happens, but why it happens. Let's walk through the exercise:

• **Step 1: Understanding the Sigma Notation**
  Sigma notation \( \sum \) is used to signify a sum. The formula \( \sum_{k=1}^{4} \sqrt{k} \) indicates that we are summing the square roots starting at \( k=1 \) and ending at \( k=4 \).
• **Step 2: Listing Each Term**
  Replace \( k \) with each integer from 1 through 4. Calculate their square roots — \( \sqrt{1}, \sqrt{2}, \sqrt{3}, \) and \( \sqrt{4} \). This breaks down the sigma notation into manageable pieces.
• **Step 3: Expanding the Sum**
  Write the expression in expanded form by adding all the terms: \( \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} \).

Using this systematic approach makes the process of solving mathematical expressions less overwhelming, turning a seemingly complex problem into an accessible sequence of simpler tasks.

A square root is a number which, when multiplied by itself, gives the original number. It's like asking, "which number squared equals this number?" In symbolic terms, the square root of \( x \) is written as \( \sqrt{x} \).Here's how it applies in our exercise:

• For \( \sqrt{1} \), since \( 1 \times 1 = 1 \), the square root is 1.
• \( \sqrt{2} \) is approximately 1.41. Perfect squares are easier, but non-perfect squares can stay as square roots in your expression.
• \( \sqrt{3} \) is about 1.73; as with \( \sqrt{2} \), this is an approximation.
• For \( \sqrt{4} \), because \( 2 \times 2 = 4 \), the square root is 2.

Understanding square roots is crucial as it helps you both decompose and solve more complex mathematical problems. They often appear in geometry and algebra, playing a key role in various equations and expressions.