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:

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

\(\displaystyle = 12.5 \times 3 - 1.2\)

\(\displaystyle = 37.5 - 1.2\)

\(\displaystyle = 36.3\)

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

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

\(\displaystyle = 20 - 7.5 \times 1.8\)

\(\displaystyle = 20 - 13.5\)

\(\displaystyle = 6.5\)

The identity property of multiplication states that there is a number 1, called the multiplicative identity, that can be multiplied by any number to obtain that number. Of the four statements, 

\(\displaystyle \frac{2}{3} \times 1 = \frac{2}{3}\)

demonstrates this property.

The product of two numbers of unlike sign is the (negative) opposite of the product of their absolute values.

\(\displaystyle 20.1 \cdot (-20.3) = - \left (20.1 \cdot 20.3 \right )\)

There are a total of two digits to the right of the two decimal points. Therefore, the product can be calculated by multiplying 201 by 203, then placing the decimal point so that there are three digits at right. 

\(\displaystyle 201 \cdot 203 = 40,803 \Rightarrow 20.1 \cdot 20.3 = 408.03\)

Since we need to affix a negative symbol in front, the answer is \(\displaystyle -408.03\)

When we multiply fractions, we multiply the numerator by the numerator and the denominator by the denominator. 

\(\displaystyle \small \small \frac{6}{7}\times\frac{1}{2}=\frac{6}{14}\)

\(\displaystyle \small \frac{6}{14}\) can be reduced by dividing both sides by \(\displaystyle \small 2\)

\(\displaystyle \small \small \frac{6}{14}\div\frac{2}{2}=\frac{3}{7}\)

When we multiply fractions, we multiply the numerator by the numerator and the denominator by the denominator. 

\(\displaystyle \small \small \frac{1}{2}\times\frac{5}{11}=\frac{5}{22}\)