Multiply the numerators together, and then multiply the denominators together:

$\displaystyle \frac{1}{3} \times \frac{4}{5}=\frac{1\times 4}{3\times 5}=\frac{4}{15}$

To solve for this expression, first multiple the numerators, and then multiply the demonators. 

$\displaystyle \frac{4}{5}\cdot\frac{7}{8}$

$\displaystyle \frac{4\times7}{5\times8}$

$\displaystyle \frac{28}{40}$

Simplify the fraction by removing a common factor.

$\displaystyle \frac{28\div4}{40\div4}=\frac{7}{10}$

To raise a negative number to an even-numbered power, raise its absolute value to that power: 

$\displaystyle (-8)^{4}= 8^{4} = 8 \times 8 \times 8 \times 8 = 4,096$

By the order of operations, carry out the operations in parentheses first; since there is a multiplication and a subtraction present, carry them out in that order. Finally, carry out the remaining subtraction:

$\displaystyle 15 - \left ( \frac{2}{3} \times 3 - \frac{4}{3}\right )$

$\displaystyle = 15 - \left ( 2- \frac{4}{3}\right )$

$\displaystyle = 15 - \frac{2}{3}$

$\displaystyle = 14 \frac{1}{3}$

The inverse property of multiplication states that for every real number, a number exists, called the multiplicative inverse, such that the number and its inverse have product 1. Of the statements given, only 

$\displaystyle \frac{6}{7} \times \frac{7}{6} = 1$

demonstrates this property.

To raise a negative number to an even-numbered power, raise its absolute value to that power. Also, to raise a fraction to a power, raise its numerator and its denominator to that power. Combine these ideas as follows:

$\displaystyle \left (- \frac{3}{2} \right )^{4} = \left( \frac{3}{2}\right )^{4} = \frac{3^{4} }{2^{4} } = \frac{ 3 \times 3\times 3 \times 3 }{ 2 \times 2 \times 2 \times 2 } \right ) = \frac{81}{16}$

To raise a negative number to an odd-numbered power, raise its absolute value to that power, then make the sign negative. Also, to raise a fraction to a power, raise its numerator and its denominator to that power. Combine these ideas as follows:

$\displaystyle \left (- \frac{2}{3} \right )^{5} = - \left ( \frac{2}{3} \right )^{5} = - \left ( \frac{2^{5} }{3^{5} } \right ) = - \left ( \frac{2\times 2 \times 2 \times 2 \times 2 }{3 \times 3 \times 3\times 3 \times 3 } \right ) = -\frac{32}{243}$

To raise a negative number to an odd-numbered power, raise its absolute value to that power, then make the sign negative: 

$\displaystyle \left ( -7\right )^{5} = - \left ( 7^{5}\right ) = -(7 \times 7 \times 7 \times 7 \times 7) = - 16,807$

One Euro is equivalent to $1.27, so multiply the number of euros - 800 - by this conversion factor.

$\displaystyle \$1.27 \times 800 = \$ 1,016$

By the order of operations, carry out the operation in parentheses, which is the leftmost subtraction, then the multiplication, then the rightmost subtraction:

$\displaystyle \left (20 - 7.5 \right )\times 3 - 1.2$

$\displaystyle = 12.5 \times 3 - 1.2$

$\displaystyle = 37.5 - 1.2$

$\displaystyle = 36.3$