Question

review sections 1.1-1.2

1. explain how to use each diagram to determine the value of the square root.

a) \\( \sqrt{\frac{1}{9}} \\)
b) \\( \sqrt{0.16} \\)

Explanation:

Step1: Analyze Diagram for \(\sqrt{\frac{1}{9}}\)

The diagram for part (a) is a square with side length 1 unit, divided into a \(3\times3\) grid (since \(3\times3 = 9\) small squares). The shaded area is 1 small square. The fraction of the shaded area is \(\frac{1}{9}\). To find the square root, we note that the side length of the shaded square (if we consider the shaded region as a square) should be such that its area is \(\frac{1}{9}\). Since the big square has side 1, each small square has side \(\frac{1}{3}\) (because \(3\) small squares along one side, so \(1\div3=\frac{1}{3}\)). The area of a square with side \(\frac{1}{3}\) is \((\frac{1}{3})^2=\frac{1}{9}\), so \(\sqrt{\frac{1}{9}}=\frac{1}{3}\).

Step2: Analyze Diagram for \(\sqrt{0.16}\)

The diagram for part (b) is a square with side length 1 unit, divided into a \(5\times5\) grid? Wait, no, let's see: \(0.16=\frac{16}{100}=\frac{4}{25}\)? Wait, no, the diagram has a shaded area of 4 small squares? Wait, the diagram is a \(5\times5\) grid? Wait, no, let's check the side length. Wait, the shaded area is 4 small squares? Wait, no, the diagram for \(\sqrt{0.16}\): \(0.16=\frac{16}{100}\), but maybe the grid is \(10\times10\)? Wait, no, the given diagram for (b) has a shaded area that is a square. Wait, the side length of the big square is 1 unit. If we divide the big square into a \(5\times5\) grid? No, wait, the answer for (b) is \(\frac{2}{5}\) as per the note. Wait, \((\frac{2}{5})^2=\frac{4}{25}=0.16\). So the diagram is a square divided into a \(5\times5\) grid (25 small squares), and the shaded area is 4 small squares? Wait, no, \(\frac{4}{25}=0.16\). So the side length of the shaded square is \(\frac{2}{5}\) (since \(\frac{2}{5}\times\frac{2}{5}=\frac{4}{25}=0.16\)). So to find \(\sqrt{0.16}\), we look at the side length of the shaded square. Since the big square has side 1, and the shaded square has side \(\frac{2}{5}\), then \(\sqrt{0.16}=\frac{2}{5}=0.4\).

Answer:

a) For \(\sqrt{\frac{1}{9}}\), the diagram is a \(3\times3\) grid (area 9 small squares) with 1 shaded square (area \(\frac{1}{9}\)). The side length of the shaded square (a smaller square within the grid) is \(\frac{1}{3}\) (since \((\frac{1}{3})^2=\frac{1}{9}\)), so \(\sqrt{\frac{1}{9}}=\frac{1}{3}\).
b) For \(\sqrt{0.16}\), the diagram is a square (side 1) with a shaded square. The shaded area is \(0.16=\frac{4}{25}\), and the side length of the shaded square is \(\frac{2}{5}\) (since \((\frac{2}{5})^2=\frac{4}{25}=0.16\)), so \(\sqrt{0.16}=\frac{2}{5}=0.4\).