Q2(iii):

Simplify each of the following expressions: (iii) $(\sqrt{5} + \sqrt{2})^2$

Given Expression & Algebraic Foundation

We are tasked with simplifying the following mathematical expression involving irrational numbers:

$(\sqrt{5} + \sqrt{2})^2$

To expand this expression systematically, we utilize the standard binomial square identity [Derived from the distributive property of multiplication over addition]:

$(a + b)^2 = a^2 + 2ab + b^2$

Step 1: Variable Substitution

By mapping the given expression to the binomial identity, we establish our variables:

• Let $a = \sqrt{5}$
• Let $b = \sqrt{2}$

Substituting these values into the identity yields:

$(\sqrt{5} + \sqrt{2})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{2}) + (\sqrt{2})^2$

Step 2: Simplifying the Squared Terms

We evaluate the first and third terms of the expansion. [Per the fundamental definition of a principal square root, $(\sqrt{x})^2 = x$ for any non-negative real number $x \ge 0$].

• First term: $(\sqrt{5})^2 = 5$
• Third term: $(\sqrt{2})^2 = 2$

Step 3: Simplifying the Cross-Product Term

Next, we evaluate the middle term, $2(\sqrt{5})(\sqrt{2})$. [Per the Product Property of Radicals, $\sqrt{x} \cdot \sqrt{y} = \sqrt{xy}$ for $x, y \ge 0$].

$2(\sqrt{5})(\sqrt{2}) = 2(\sqrt{5 \cdot 2}) = 2\sqrt{10}$

Step 4: Combining Like Terms

Substituting the simplified components back into the expanded equation, we get:

$5 + 2\sqrt{10} + 2$

We group the rational numbers together and leave the irrational term distinct [Per the commutative property of addition]:

$(5 + 2) + 2\sqrt{10} = 7 + 2\sqrt{10}$

Geometric Visualization of the Expansion

The algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$ can be visualized as the area of a square with side length $(a+b)$, partitioned into four distinct rectangular regions. Below is the geometric proof applied specifically to $a = \sqrt{5}$ and $b = \sqrt{2}$.

Summing the areas of the four regions confirms our algebraic derivation:

$\text{Total Area} = 5 + \sqrt{10} + \sqrt{10} + 2 = 7 + 2\sqrt{10}$

Final Solution: $7 + 2\sqrt{10}$