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The perimeter of a rectangle is the total distance around the outer edge. To find it, you use the perimeter formula: \( P = 2(l + w) \). Here, \( l \) represents the length and \( w \) the width. This formula makes sense because you are essentially adding up all four sides of the rectangle: two lengths and two widths.

• First, you calculate \( l + w \) (add the length and width together)
• Next, multiply the sum by 2, reflecting the two pairs of sides in a rectangle

So, for our exercise, we substitute the given length and width into the formula: \( P = 2(26\sqrt{\text{ cm}} + 3\sqrt{\text{ cm}}) \). Simplify inside the parentheses first to get \( 29\sqrt{\text{ cm}} \). Then, multiply this by 2 to get the final perimeter: \( 58\sqrt{\text{ cm}} \). This shows that even with complex numbers like these involving square roots, the formula remains straightforward to apply.

The area of a rectangle is a measure of the total space inside the shape. It's calculated with the formula: \( A = l \times w \), where you multiply the length by the width. For a rectangular surface, this formula is helpful in determining how much material might cover it, how much space it takes up, or how much paint you might need.

• Identify the length \( l \) and width \( w \)
• Multiply them together to get the area

In this example, we replace \( l \) and \( w \) with their values: \( A = 26\sqrt{\text{ cm}} \times 3\sqrt{\text{ cm}} \). To simplify:

• Multiply the coefficients: \( 26 \times 3 = 78 \)
• Square the units \( (\sqrt{\text{ cm}})^2 = \text{ cm} \)

Thus, the area is \( 78 \text{ cm} \). Remember: the area gives us a two-dimensional measure, unlike the perimeter which is linear.

Simplifying mathematical expressions, especially those involving square roots, is an essential math skill. When dealing with expressions like \( 26\sqrt{\text{ cm}} + 3\sqrt{\text{ cm}} \), it's crucial to treat it as combining like terms. These terms are similar because they both involve \( \sqrt{\text{ cm}} \), allowing us to add them directly. Here's how it works:

• Identify the like terms (terms that share the same base variables or units)
• Combine them by adding or subtracting their coefficients

For instance, \( 26\sqrt{\text{ cm}} + 3\sqrt{\text{ cm}} = 29\sqrt{\text{ cm}} \). Breaking it down helps manage more complex calculations effectively. Also, simplifying products, like \( 26\sqrt{\text{ cm}} \times 3\sqrt{\text{ cm}} \), involves:

• Calculating the product of the numbers: \( 26 \times 3 = 78 \)
• Handling the square roots separately: \((\sqrt{\text{ cm}})^2 = \text{ cm} \)

Such techniques make solving rectangle area and perimeter problems less daunting.