Distribute sqrt(5) to each term inside the parenthesis.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a)
Step 1: Distribute $\sqrt{5}$ to each term inside the parenthesis.
$$ \sqrt{5}(\sqrt{2} + 1) = \sqrt{5} \times \sqrt{2} + \sqrt{5} \times 1 $$
Step 2: Multiply the surds and simplify.
$$ \sqrt{5 \times 2} + \sqrt{5} = \sqrt{10} + \sqrt{5} $$
The simplified expression is:
$$ \boxed{\sqrt{10} + \sqrt{5}} $$
b)
Step 1: Distribute $\sqrt{5}$ to each term inside the parenthesis.
$$ \sqrt{5}(3 - \sqrt{2}) = \sqrt{5} \times 3 - \sqrt{5} \times \sqrt{2} $$
Step 2: Multiply the terms.
$$ 3\sqrt{5} - \sqrt{5 \times 2} = 3\sqrt{5} - \sqrt{10} $$
The simplified expression is:
$$ \boxed{3\sqrt{5} - \sqrt{10}} $$
c)
Step 1: Apply the exponent to both factors inside the parenthesis.
$$ (2\sqrt{2})^3 = 2^3 \times (\sqrt{2})^3 $$
Step 2: Calculate $2^3$.
$$ 2^3 = 2 \times 2 \times 2 = 8 $$
Step 3: Calculate $(\sqrt{2})^3$.
$$ (\sqrt{2})^3 = \sqrt{2} \times \sqrt{2} \times \sqrt{2} = (\sqrt{2})^2 \times \sqrt{2} = 2 \times \sqrt{2} = 2\sqrt{2} $$
Step 4: Multiply the results from Step 2 and Step 3.
$$ 8 \times 2\sqrt{2} = 16\sqrt{2} $$
The simplified expression is:
$$ \boxed{16\sqrt{2}} $$
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