Question

simplify square root expressions
part i

1. in determining how many bags of fertilizer we need, what is the first calculation we need to make?
2. complete the table to find the area of each of the 3 different gardens

	radish	garden 1	garden 2
expanded form	$2 \cdot \sqrt{2} \cdot 2 \cdot \sqrt{2}$
coefficients multiplied	$4 \cdot \sqrt{2} \cdot \sqrt{2}$
radicands multiplied	$4 \cdot \sqrt{4}$
radicals simplified	$4 \cdot 2$
area expression simplified	8

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Explanation:

Step1: Expand the expression for Garden 1

The area expression for Garden 1 is \(5\sqrt{6} \cdot 10\sqrt{2}\). Expanding it, we separate the coefficients and the radicals: \(5 \cdot \sqrt{6} \cdot 10 \cdot \sqrt{2}\)

Step2: Multiply the coefficients

Multiply the coefficients \(5\) and \(10\): \(5\times10 = 50\), so we have \(50 \cdot \sqrt{6} \cdot \sqrt{2}\)

Step3: Multiply the radicands

Multiply the radicands \(\sqrt{6}\) and \(\sqrt{2}\) using the property \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\), so \(\sqrt{6}\cdot\sqrt{2}=\sqrt{6\times2}=\sqrt{12}\). Now we have \(50 \cdot \sqrt{12}\)

Step4: Simplify the radical

Simplify \(\sqrt{12}\). We know that \(12 = 4\times3\), so \(\sqrt{12}=\sqrt{4\times3}=\sqrt{4}\cdot\sqrt{3} = 2\sqrt{3}\). Substituting back, we get \(50\times2\sqrt{3}\)

Step5: Multiply the coefficient and the simplified radical

Multiply \(50\) and \(2\): \(50\times2 = 100\), so the simplified area expression is \(100\sqrt{3}\)

Now for Garden 2:

Step1: Expand the expression for Garden 2

The area expression for Garden 2 is \(10\sqrt{2} \cdot 4\sqrt{6}\). Expanding it, we separate the coefficients and the radicals: \(10 \cdot \sqrt{2} \cdot 4 \cdot \sqrt{6}\)

Step2: Multiply the coefficients

Multiply the coefficients \(10\) and \(4\): \(10\times4 = 40\), so we have \(40 \cdot \sqrt{2} \cdot \sqrt{6}\)

Step3: Multiply the radicands

Multiply the radicands \(\sqrt{2}\) and \(\sqrt{6}\) using the property \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\), so \(\sqrt{2}\cdot\sqrt{6}=\sqrt{2\times6}=\sqrt{12}\). Now we have \(40 \cdot \sqrt{12}\)

Step4: Simplify the radical

Simplify \(\sqrt{12}\). We know that \(12 = 4\times3\), so \(\sqrt{12}=\sqrt{4\times3}=\sqrt{4}\cdot\sqrt{3} = 2\sqrt{3}\). Substituting back, we get \(40\times2\sqrt{3}\)

Step5: Multiply the coefficient and the simplified radical

Multiply \(40\) and \(2\): \(40\times2 = 80\), so the simplified area expression is \(80\sqrt{3}\)

Filling the table:

	Radish	Garden 1	Garden 2
Expanded Form	\(2\cdot\sqrt{2}\cdot2\cdot\sqrt{2}\)	\(5\cdot\sqrt{6}\cdot10\cdot\sqrt{2}\)	\(10\cdot\sqrt{2}\cdot4\cdot\sqrt{6}\)
Coefficients Multiplied	\(4\cdot\sqrt{2}\cdot\sqrt{2}\)	\(50\cdot\sqrt{6}\cdot\sqrt{2}\)	\(40\cdot\sqrt{2}\cdot\sqrt{6}\)
Radicands Multiplied	\(4\cdot\sqrt{4}\)	\(50\cdot\sqrt{12}\)	\(40\cdot\sqrt{12}\)
Radicals Simplified	\(4\cdot2\)	\(50\cdot2\sqrt{3}\)	\(40\cdot2\sqrt{3}\)
Area Expression Simplified	\(8\)	\(100\sqrt{3}\)	\(80\sqrt{3}\)

Answer:

For Garden 1, the simplified area expression is \(100\sqrt{3}\) and for Garden 2, it is \(80\sqrt{3}\) (and the table is filled as above)