Square-root the whole area • Pull out perfect-square factors • Check by squaring
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Area to side
Square area means both sides are equal, so invert by taking one square root of the whole area.
$$s = \sqrt{A}$$
Product under a radical
Split only into two square roots. Perfect-square factors come out; everything else stays inside.
$$\sqrt{4\pi} = \sqrt{4} \cdot \sqrt{\pi} = 2\sqrt{\pi}$$
Always square to check
The side is correct only if its square is exactly the given area.
$$(2\sqrt{\pi})^2 = 4\pi$$
But \((2\pi)^2 = 4\pi^2\)
Work in order. Problem 1 rebuilds “side = square root of area.” Problem 2 uses the same algebra with an ordinary number instead of \(\pi\). Problem 3 isolates how \(\pi\) behaves under a radical. Problem 4 applies the product rule to \(25\pi\). Problem 5 transfers the skill to a new coefficient.
A square has an area of 36. What is the side length?
Area of a square: \(s^2 = 36\).
$$s = \sqrt{36} = 6$$
Check: \(6^2 = 36\). (Length is positive, so we take the principal square root.)
Answer: \(6\)
A square has an area of 20. Write the side length in simplest radical form.
$$s = \sqrt{20} = \sqrt{4\cdot 5} = \sqrt{4}\cdot\sqrt{5} = 2\sqrt{5}$$
Common wrong move: writing \(2\cdot 5 = 10\). That would be “square-root the \(4\) and leave the \(5\) outside,” which is the same error as writing \(2\pi\) for \(\sqrt{4\pi}\).
Check: \((2\sqrt{5})^2 = 4\cdot 5 = 20\).
Answer: \(2\sqrt{5}\)
Simplify each expression. Why are the two answers different?
(a) \(\sqrt{9\pi}\) (b) \(\sqrt{9\pi^2}\)
(a) \(9\) is a perfect square. \(\pi\) is not.
$$\sqrt{9\pi} = \sqrt{9}\cdot\sqrt{\pi} = 3\sqrt{\pi}$$
(b) Both \(9\) and \(\pi^2\) are perfect squares.
$$\sqrt{9\pi^2} = \sqrt{9}\cdot\sqrt{\pi^2} = 3\pi$$
\(\pi\) comes out of the radical only when it is squared (or has an even power). A lone \(\pi\) stays inside.
Answer: \(3\sqrt{\pi}\) and \(3\pi\)
A square has an area of 25π.
(a) Find the exact side length. (b) A classmate says the side is 5π. Show that this cannot be correct by squaring both answers.
(a)
$$s = \sqrt{25\pi} = \sqrt{25}\cdot\sqrt{\pi} = 5\sqrt{\pi}$$
(b) Square each candidate and compare to the given area \(25\pi\):
$$\left(5\sqrt{\pi}\right)^2 = 5^2 \cdot \left(\sqrt{\pi}\right)^2 = 25\pi \quad \text{matches the area}$$
$$\left(5\pi\right)^2 = 25\pi^2 \quad \text{too large by a factor of }\pi$$
\(5\) and \(5\pi\) are a factor pair of \(25\pi\), so they could be the length and width of a rectangle — not the equal sides of a square.
Answer: \(5\sqrt{\pi}\); \(5\pi\) is wrong because \((5\pi)^2 = 25\pi^2 \neq 25\pi\)
A square has an area of 18π. Write the side length in simplest radical form.
Then decide which is larger: the side of this square, or 3π.
$$s = \sqrt{18\pi} = \sqrt{9\cdot 2\pi} = 3\sqrt{2\pi}$$
Equivalent form: \(3\sqrt{2}\sqrt{\pi}\). Do not write \(3\pi\) or \(9\pi\).
Compare \(3\sqrt{2\pi}\) with \(3\pi\) by squaring (both positive):
$$\left(3\sqrt{2\pi}\right)^2 = 9\cdot 2\pi = 18\pi$$
$$(3\pi)^2 = 9\pi^2$$
Since \(9\pi^2 > 18\pi\) (\(\pi > 2\)), we have \(3\pi > 3\sqrt{2\pi}\). The side is smaller than \(3\pi\).
Answer: \(3\sqrt{2\pi}\); \(3\pi\) is larger
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