Question

do these problems on some clean paper. label each page of your work with your name, your class, the date and the book number. also number each problem. keep this written work inside your book, and turn it in with your book when you are finished. please do a neat job. 1. explain why 1.4142135 ≈ √2. 2. copy each expression which stands for a rational number and tell what rational number it equals. √5 √25 √20 √20√5 √5√5 √20/√5 √25/√5 √25/√4 -√5 -√4 √-4 √-1 √0 √1 √10 √100 √1000 √10000 3. simplify each expression. √50 √3 √(1/7) √(5/9) √(5/6) 2√12 3/√2 √6x² √49x √24 + √600

Explanation:

1. Explanation why \(1.4142135\approx\sqrt{2}\)

Step1: Square the approximation

Calculate \(1.4142135^2\).
\[1.4142135\times1.4142135 = 1.99999992146225\approx2\]

Step2: Recall square - root definition

Since the square - root of a number \(x\) is a value \(y\) such that \(y^2=x\), and \(1.4142135^2\approx2\), we can say \(1.4142135\approx\sqrt{2}\).

Step1: Recall rational number definition

A rational number is a number that can be written as \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q
eq0\). Also, if the number under the square - root is a perfect square, its square - root is rational.

Step2: Analyze each expression

• \(\sqrt{5}\): Irrational since 5 is not a perfect square.
• \(\sqrt{25}=5\) (rational, because \(5=\frac{5}{1}\)).
• \(\sqrt{20}=2\sqrt{5}\) (irrational).
• \(\sqrt{20}\sqrt{5}=\sqrt{100} = 10\) (rational, because \(10=\frac{10}{1}\)).
• \(\sqrt{5}\sqrt{5}=5\) (rational, because \(5=\frac{5}{1}\)).
• \(\frac{\sqrt{20}}{\sqrt{5}}=\sqrt{4} = 2\) (rational, because \(2=\frac{2}{1}\)).
• \(\frac{\sqrt{25}}{\sqrt{5}}=\sqrt{5}\) (irrational).
• \(\frac{\sqrt{25}}{\sqrt{4}}=\frac{5}{2}\) (rational).
• \(-\sqrt{5}\): Irrational.
• \(-\sqrt{4}=-2\) (rational, because \(-2=\frac{-2}{1}\)).
• \(\sqrt{-4}\) and \(\sqrt{-1}\): Not real numbers (complex numbers).
• \(\sqrt{0}=0\) (rational, because \(0=\frac{0}{1}\)).
• \(\sqrt{1}=1\) (rational, because \(1=\frac{1}{1}\)).
• \(\sqrt{10}\): Irrational since 10 is not a perfect square.
• \(\sqrt{100}=10\) (rational, because \(10=\frac{10}{1}\)).
• \(\sqrt{1000}=10\sqrt{10}\) (irrational).
• \(\sqrt{10000}=100\) (rational, because \(100=\frac{100}{1}\)).

Step1: For \(\sqrt{50}\)

Factor 50 as \(25\times2\), then \(\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}\).

Step2: For \(\sqrt{3}\)

It is already in simplest form since 3 is a prime number.

Step3: For \(\sqrt{\frac{1}{7}}\)

Rationalize the denominator: \(\sqrt{\frac{1}{7}}=\frac{\sqrt{1}}{\sqrt{7}}=\frac{1}{\sqrt{7}}=\frac{\sqrt{7}}{7}\).

Step4: For \(\sqrt{\frac{5}{9}}\)

\(\sqrt{\frac{5}{9}}=\frac{\sqrt{5}}{\sqrt{9}}=\frac{\sqrt{5}}{3}\).

Step5: For \(\sqrt{\frac{5}{6}}\)

Rationalize the denominator: \(\sqrt{\frac{5}{6}}=\frac{\sqrt{5}}{\sqrt{6}}=\frac{\sqrt{5}\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}=\frac{\sqrt{30}}{6}\).

Step6: For \(2\sqrt{12}\)

Factor 12 as \(4\times3\), then \(2\sqrt{12}=2\sqrt{4\times3}=2\times\sqrt{4}\times\sqrt{3}=2\times2\times\sqrt{3}=4\sqrt{3}\).

Step7: For \(\frac{3}{\sqrt{2}}\)

Rationalize the denominator: \(\frac{3}{\sqrt{2}}=\frac{3\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{3\sqrt{2}}{2}\).

Step8: For \(\sqrt{6x^{2}}\)

\(\sqrt{6x^{2}}=\vert x\vert\sqrt{6}\).

Step9: For \(\sqrt{49x}=7\sqrt{x}\) (assuming \(x\geq0\)).

Step10: For \(\sqrt{24}+\sqrt{600}\)

Factor 24 as \(4\times6\) and 600 as \(100\times6\). Then \(\sqrt{24}+\sqrt{600}=\sqrt{4\times6}+\sqrt{100\times6}=2\sqrt{6}+10\sqrt{6}=12\sqrt{6}\).

Answer:

Because \(1.4142135^2\approx2\) and by the definition of square - root (\(y = \sqrt{x}\) means \(y^2=x\)), \(1.4142135\approx\sqrt{2}\).

2. Identifying rational numbers among square - root expressions