8. Write each number in the form $$ \frac{a}{b} $$ where $$ a $$ and $$ b \in \mathbb{Z} $$ and $$ a $$ and $$ b $$ have no common factors. $$ \begin{array}{ll} ext { a) } \frac{2}{3} imes \frac{6}{5} & ext { b) } \frac{-15}{8} imes \frac{12}{10} \ ext { c) }-1 \frac{3}{4} imes \frac{8}{9} & ext { d) } 2 \frac{4}{5} imes \frac{-10}{12} \ ext { e) } \frac{4}{6} \div \frac{4}{3} & ext { f) } \frac{-6}{25} \div \frac{5}{12} \ ext { g) } \frac{1}{2}+\frac{1}{3}+\frac{1}{4} & ext { h) } 2 \frac{2}{3}+1 \frac{1}{4}-3 \frac{1}{5} \ ext { i) } \frac{2}{3} imes \frac{3}{4} \div \frac{5}{4}+\frac{4}{5} \div \frac{1}{2} & ext { j) } \frac{2}{3} \div \frac{3}{4} \div \frac{5}{4}+\frac{4}{5} \div 2 \ ext { k) } \frac{2}{3} \div\left(\frac{3}{4} \div \frac{5}{4} ight)-\frac{4}{3} \div \frac{6}{2} & ext { l) } \frac{3}{2} \div \frac{3}{4} imes\left(\frac{5}{4}-\frac{4}{5} ight) \div \frac{1}{2} \ ext { Use your calculator to find an approximate value for } \sqrt{7} \ ext { Is } \sqrt{7} ext { a Rational or an Irrational Number? } \ \begin{array}{ll} ext { Determine the two consecutive Integers that each Irrational } \ ext { Number lies between: } & ext { l. } \ ext { a) } \sqrt{50} & ext { if } \ ext { b) } \sqrt[3]{45} & \end{array}\end{array} \begin{array}{l} ext { c) } 2 \sqrt{6}\end{array} $$

Question

8. Write each number in the form $$ \frac{a}{b} $$ where $$ a $$ and $$ b \in \mathbb{Z} $$ and $$ a $$ and $$ b $$ have no common factors. $$ \begin{array}{ll}	ext { a) } \frac{2}{3} 	imes \frac{6}{5} & 	ext { b) } \frac{-15}{8} 	imes \frac{12}{10} \ 	ext { c) }-1 \frac{3}{4} 	imes \frac{8}{9} & 	ext { d) } 2 \frac{4}{5} 	imes \frac{-10}{12} \ 	ext { e) } \frac{4}{6} \div \frac{4}{3} & 	ext { f) } \frac{-6}{25} \div \frac{5}{12} \ 	ext { g) } \frac{1}{2}+\frac{1}{3}+\frac{1}{4} & 	ext { h) } 2 \frac{2}{3}+1 \frac{1}{4}-3 \frac{1}{5} \ 	ext { i) } \frac{2}{3} 	imes \frac{3}{4} \div \frac{5}{4}+\frac{4}{5} \div \frac{1}{2} & 	ext { j) } \frac{2}{3} \div \frac{3}{4} \div \frac{5}{4}+\frac{4}{5} \div 2 \ 	ext { k) } \frac{2}{3} \div\left(\frac{3}{4} \div \frac{5}{4}ight)-\frac{4}{3} \div \frac{6}{2} & 	ext { l) } \frac{3}{2} \div \frac{3}{4} 	imes\left(\frac{5}{4}-\frac{4}{5}ight) \div \frac{1}{2} \ 	ext { Use your calculator to find an approximate value for } \sqrt{7} \ 	ext { Is } \sqrt{7} 	ext { a Rational or an Irrational Number? } \ \begin{array}{ll}	ext { Determine the two consecutive Integers that each Irrational } \ 	ext { Number lies between: } & 	ext { l. } \ 	ext { a) } \sqrt{50} & 	ext { if } \ 	ext { b) } \sqrt[3]{45} & \end{array}\end{array} \begin{array}{l}	ext { c) } 2 \sqrt{6}\end{array} $$

UpStudy AI · Accepted Answer

Let's solve each part step by step, ensuring that we express each number in the form $$ \frac{a}{b} $$ where $$ a $$ and $$ b $$ are integers with no common factors.

### a) $$ \frac{2}{3} \times \frac{6}{5} $$

$$
\frac{2}{3} \times \frac{6}{5} = \frac{2 \times 6}{3 \times 5} = \frac{12}{15}
$$

Now, we simplify $$ \frac{12}{15} $$:

The GCD of 12 and 15 is 3.

$$
\frac{12 \div 3}{15 \div 3} = \frac{4}{5}
$$

### b) $$ \frac{-15}{8} \times \frac{12}{10} $$

$$
\frac{-15}{8} \times \frac{12}{10} = \frac{-15 \times 12}{8 \times 10} = \frac{-180}{80}
$$

Now, we simplify $$ \frac{-180}{80} $$:

The GCD of 180 and 80 is 20.

$$
\frac{-180 \div 20}{80 \div 20} = \frac{-9}{4}
$$

### c) $$ -1 \frac{3}{4} \times \frac{8}{9} $$

First, convert $$ -1 \frac{3}{4} $$ to an improper fraction:

$$
-1 \frac{3}{4} = -\frac{4 + 3}{4} = -\frac{7}{4}
$$

Now, multiply:

$$
-\frac{7}{4} \times \frac{8}{9} = \frac{-7 \times 8}{4 \times 9} = \frac{-56}{36}
$$

Now, we simplify $$ \frac{-56}{36} $$:

The GCD of 56 and 36 is 4.

$$
\frac{-56 \div 4}{36 \div 4} = \frac{-14}{9}
$$

### d) $$ 2 \frac{4}{5} \times \frac{-10}{12} $$

Convert $$ 2 \frac{4}{5} $$ to an improper fraction:

$$
2 \frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{10 + 4}{5} = \frac{14}{5}
$$

Now, multiply:

$$
\frac{14}{5} \times \frac{-10}{12} = \frac{14 \times -10}{5 \times 12} = \frac{-140}{60}
$$

Now, we simplify $$ \frac{-140}{60} $$:

The GCD of 140 and 60 is 20.

$$
\frac{-140 \div 20}{60 \div 20} = \frac{-7}{3}
$$

### e) $$ \frac{4}{6} \div \frac{4}{3} $$

Dividing by a fraction is the same as multiplying by its reciprocal:

$$
\frac{4}{6} \div \frac{4}{3} = \frac{4}{6} \times \frac{3}{4} = \frac{4 \times 3}{6 \times 4} = \frac{12}{24}
$$

Now, we simplify $$ \frac{12}{24} $$:

The GCD of 12 and 24 is 12.

$$
\frac{12 \div 12}{24 \div 12} = \frac{1}{2}
$$

### f) $$ \frac{-6}{25} \div \frac{5}{12} $$

$$
\frac{-6}{25} \div \frac{5}{12} = \frac{-6}{25} \times \frac{12}{5} = \frac{-6 \times 12}{25 \times 5} = \frac{-72}{125}
$$

This fraction is already in simplest form.

### g) $$ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} $$

To add these fractions, we need a common denominator. The least common multiple of 2, 3, and 4 is 12.

$$
\frac{1}{2} = \frac{6}{12}, \quad \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}
$$

Now, add them:

$$
\frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{6 + 4 + 3}{12} = \frac{13}{12}
$$

### h) $$ 2 \frac{2}{3} + 1 \frac{1}{4} - 3 \frac{1}{5} $$

Convert to improper fractions:

$$
2 \frac{2}{3} = \frac{8}{3}, \quad 1 \frac{1}{4} = \frac{5}{4}, \quad 3 \frac{1}{5} = \frac{16}{5}
$$

Finding a common denominator (60):

$$
\frac{8}{3} = \frac{160}{60}, \quad \frac{5}{4} = \frac{75}{60}, \quad \frac{16}{5} = \frac{192}{60}
$$

Now, combine:

$$
\frac{160}{60} + \frac{75}{60} - \frac{192}{60} = \frac{160 + 75 - 192}{60} = \frac{43}{60}
$$

### i) $$ \frac{2}{3} \times \frac{3}{4} \div \frac{5}{4} + \frac{4}{5} \div \frac{1}{2} $$

First, calculate $$ \frac{2}{3} \times \frac{3}{4} $$:

$$
\frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}
$$

Now divide by $$ \frac{5}{4} $$:

$$
\frac{1}{2} \div \frac{5}{4} = \frac{1}{2} \times \frac{4}{5} = \frac{4}{10} = \frac{2}{5}
$$

Now calculate $$ \frac{4}{5} \div \frac{1}{2} $$:

$$
\frac{4}{5} \times 2 = \frac{8}{5}
$$

Now add:

\[
\frac{2}{5}
a) $$ \frac{4}{5} $$ b) $$ \frac{-9}{4} $$ c) $$ \frac{-14}{9} $$ d) $$ \frac{-7}{3} $$ e) $$ \frac{1}{2} $$ f) $$ \frac{-72}{125} $$ g) $$ \frac{13}{12} $$ h) $$ \frac{43}{60} $$ i) $$ \frac{2}{5} $$ j) $$ \frac{2}{5} $$ k) $$ \frac{2}{5} $$ l) $$ \frac{3}{2} \div \frac{3}{4} \times \left( \frac{5}{4} - \frac{4}{5} \right) \div \frac{1}{2} = \frac{3}{2} \times \frac{4}{3} \times \left( \frac{25}{20} - \frac{16}{20} \right) \times 2 = 2 \times \frac{9}{20} \times 2 = \frac{36}{20} = \frac{9}{5} $$ Use your calculator to find an approximate value for $$ \sqrt{7} $$: $$ \sqrt{7} \approx 2.6458 $$ Is $$ \sqrt{7} $$ a Rational or an Irrational Number? $$ \sqrt{7} $$ is an irrational number. Determine the two consecutive Integers that each Irrational Number lies between: a) $$ \sqrt{50} $$ $$ \sqrt{49} = 7 $$ and $$ \sqrt{64} = 8 $$, so $$ \sqrt{50} $$ lies between 7 and 8. b) $$ \sqrt[3]{45} $$ $$ \sqrt[3]{64} = 4 $$ and $$ \sqrt[3]{125} = 5 $$, so $$ \sqrt[3]{45} $$ lies between 3 and 4. c) $$ 2 \sqrt{6} $$ $$ \sqrt{4} = 2 $$ and $$ \sqrt{9} = 3 $$, so $$ 2 \sqrt{6} $$ lies between 4 and 6.

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