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Divide the following fractions: $\frac{\frac{4}{9}}{\frac{8}{7}} \div \frac{3}{\frac{2}{3}}$

$\frac{7}{81}$

$\frac{64}{567}$

$\frac{4}{7}$

$\frac{9}{7}$

$\frac{19}{81}$

Explanation

In order to solve this, we will need to evaluate term by term. First rewrite the complex fractions by using a division sign.

$\frac{\frac{4}{9}}{\frac{8}{7}} =\frac{4}{9} \div\frac{8}{7}$

Change the sign from a division to multiplication and take the reciprocal of the second term.

$\frac{4}{9} \div\frac{8}{7} = \frac{4}{9} \times\frac{7}{8} = \frac{28}{72} =\frac{7\times 4}{18\times 4}=\frac{7}{18}$

Evaluate the second complex fraction.

$\frac{3}{\frac{2}{3}} = 3\div \frac{2}{3} = 3\times \frac{3}{2} = \frac{9}{2}$

This means that:

$\frac{\frac{4}{9}}{\frac{8}{7}} \div \frac{3}{\frac{2}{3}} = \frac{7}{18}\div \frac{9}{2}=\frac{7}{18}\times \frac{2}{9} = \frac{7}{9}\times\frac{1}{9}$

The answer is: $\frac{7}{81}$

Divide the fractions: $\frac{10}{3}\div\frac{7}{9}$

$\frac{30}{7}$

$\frac{70}{27}$

$\frac{27}{70}$

$\frac{7}{30}$

$\frac{17}{3}$

Explanation

Change the division sign in the expression and take the reciprocal of the second term.

$\frac{10}{3}\div\frac{7}{9} = \frac{10}{3}\times\frac{9}{7}$

Reduce the three and nine in the numerator and denominator.

The fractions become:

$\frac{10}{1} \times \frac{3}{7} = \frac{30}{7}$

The answer is: $\frac{30}{7}$

Divide the fractions: $\frac{\frac{2}{3}}{5}\div \frac{\frac{1}{2}}{3}$

$\frac{4}{5}$

$\frac{5}{4}$

$\frac{1}{45}$

$\textup{The answer is not given.}$

Explanation

We will need to solve each complex fraction first.

Rewrite the complex fractions using a division sign, change the sign to a multiplication sign, and then take the reciprocal of the second term.

$\frac{\frac{2}{3}}{5} = \frac{2}{3}\div 5 =\frac{2}{3}\times \frac{1}{5} = \frac{2}{15}$

$\frac{\frac{1}{2}}{3} = \frac{1}{2}\div 3 = \frac{1}{2}\times \frac{1}{3} = \frac{1}{6}$

Divide the two fractions.

$\frac{2}{15}\div \frac{1}{6} = \frac{2}{15}\times \frac{6}{1} = \frac{12}{15}$

Reduce this fraction.

The answer is: $\frac{4}{5}$

Multiply the fractions:

$\frac{2}{5}\cdot\frac{2}{3}$

$\frac{4}{15}$

$\frac{6}{15}$

$\frac{1}{5}$

$\frac{8}{15}$

Explanation

When multiplying fractions, multiply the numbers on the top together and the numbers on the bottom together. Then simplify accordingly.

$\frac{2}{5}\cdot \frac{2}{3}=\frac{4}{15}$

Divide these fractions:

$\frac{25}{60}\div \frac{1}{5}$

$2\frac{1}{12}$

$1\frac{1}{12}$

$2\frac{3}{12}$

Explanation

When dividing fractions, first we need to flip the second fraction. Then multiply the numerators together and multiply the denominators together:

$\frac{25}{60}\div \frac{1}{5}=\frac{25}{60}*\frac{5}{1}=\frac{25*5}{60}=\frac{125}{60}$

Simplify the fraction to get the final answer:

$\frac{125}{60}\div \frac{5}{5}=\frac{25}{12}=2\frac{1}{12}$

Evaluate: $\frac{i^6+1}{i-1}$

$\frac{1}{2}i-\frac{1}{2}$

Explanation

The imaginary term is equivalent to $\sqrt{-1}$ .

This means that:

Substitute this term back into the numerator.

$\frac{i^6+1}{i-1}=\frac{-1+1}{i-1} = \frac{0}{i-1} =0$

There is no need to use extra steps such as multiplying by the conjugate of the denominator to simplify.

The answer is:

Simplify: $\frac{i^{60}-1}{i^3-1}$

$\textup{Undefined.}$

Explanation

Write the first few terms of the imaginary term.

$i=\sqrt{-1}$

$i^2 = i\cdot i =-1$

$i^3 = i^2\cdot i = -i$

$i^4 = i^2\cdot i^2=1$

Notice that these terms will be in a pattern for higher order imaginary terms.

Rewrite the numerator using the product of exponents.

$\frac{i^{60}-1}{i^3-1} =\frac{(i^4)^{15}-1}{i^3-1} = \frac{1-1}{-i-1} = 0$

The answer is:

Evaluate:

$i^{73 }$

Explanation

To raise to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the table below.

Powers of i

$73 \div 4 = 18 \textrm{ R }1$ , so $i^{73 }$ can be determined by selecting the power of corresponding to remainder 1. The correct power is , so $i^{73 } =i$ .

Compute: $-5i^{1234}$

Explanation

Identify the first two powers of the imaginary term.

$i=\sqrt{-1}$

$i^2=i\cdot i =-1$

Rewrite the expression as a product of exponents.

$-5i^{1234}=-5(i^{2})^{617} = -5(-1)^{617}$

Negative one to an odd power will be negative one.

The answer is:

Evaluate:

$| 9- i\sqrt{6}|$

$\sqrt{87}$

$5 \sqrt{3}$

$9+ \sqrt{6}$

$9- \sqrt{6}$

None of these

Explanation

refers to the absolute value of a complex number , which can be calculated by evaluating $\sqrt{a^{2}+ b^{2}}$ . Setting $a= 9, b = \sqrt{6}$ , the value of this expression is

$| 9- i\sqrt{6}| = \sqrt{9^{2}+ (\sqrt{6})^{2}} = \sqrt{81+ 6} = \sqrt{87}$

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