Simplify: a) √20 b) (√5)^2 c) 8/√2
Understand the Problem
The question asks to simplify expressions involving square roots. This requires understanding how to factor numbers to extract perfect squares from radicals, how squaring a square root cancels out, and how to rationalize denominators.
Answer
a) $2\sqrt{5}$ b) $5$ c) $4\sqrt{2}$
Answer for screen readers
a) $2\sqrt{5}$ b) $5$ c) $4\sqrt{2}$
Steps to Solve
- Simplify $\sqrt{20}$
We need to find the largest perfect square that divides 20. The largest perfect square factor of 20 is 4, since $20 = 4 \times 5$. Therefore, we can rewrite the expression as:
$\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$
- Simplify $(\sqrt{5})^2$
Squaring a square root cancels out the radical.
$(\sqrt{5})^2 = 5$
- Simplify $\frac{8}{\sqrt{2}}$
To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{2}$:
$\frac{8}{\sqrt{2}} = \frac{8}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}$
a) $2\sqrt{5}$ b) $5$ c) $4\sqrt{2}$
More Information
Rationalizing the denominator is a standard practice in simplifying radical expressions to remove any square roots from the denominator.
Tips
A common mistake is not simplifying the radical completely in part a. For example, writing $\sqrt{20} = \sqrt{4 \cdot 5}$ is a good start, but you need to continue and write $\sqrt{4 \cdot 5} = 2\sqrt{5}$. Another common mistake is forgetting to rationalize the denominator completely, or making arithmetic errors while doing so.
AI-generated content may contain errors. Please verify critical information
