Solve (1)(8/27)^-frac{2{3}}+(2/5)^-2-(-1/3+sqrt{5})^0

$$( 1 ) ( \frac { 8 } { 27 } ) ^ { - \frac { 2 } { 3 } } + ( \frac { 2 } { 5 } ) ^ { - 2 } - ( - \frac { 1 } { 3 + \sqrt { 5 } } ) ^ { 0 }$$

Evaluate

$$\frac{15}{2}=7.5$$

Solution Steps

Calculate \frac{8}{27} to the power of -\frac{2}{3} and get \frac{9}{4}.

$$1\times \frac{9}{4}+\left(\frac{2}{5}\right)^{-2}-\left(-\frac{1}{3+\sqrt{5}}\right)^{0}$$

Multiply 1 and \frac{9}{4} to get \frac{9}{4}.

$$\frac{9}{4}+\left(\frac{2}{5}\right)^{-2}-\left(-\frac{1}{3+\sqrt{5}}\right)^{0}$$

Calculate \frac{2}{5} to the power of -2 and get \frac{25}{4}.

$$\frac{9}{4}+\frac{25}{4}-\left(-\frac{1}{3+\sqrt{5}}\right)^{0}$$

Add \frac{9}{4} and \frac{25}{4} to get \frac{17}{2}.

$$\frac{17}{2}-\left(-\frac{1}{3+\sqrt{5}}\right)^{0}$$

Rationalize the denominator of \frac{1}{3+\sqrt{5}} by multiplying numerator and denominator by 3-\sqrt{5}.

$$\frac{17}{2}-\left(-\frac{3-\sqrt{5}}{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}\right)^{0}$$

Consider \left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{17}{2}-\left(-\frac{3-\sqrt{5}}{3^{2}-\left(\sqrt{5}\right)^{2}}\right)^{0}$$

Square 3. Square \sqrt{5}.

$$\frac{17}{2}-\left(-\frac{3-\sqrt{5}}{9-5}\right)^{0}$$

Subtract 5 from 9 to get 4.

$$\frac{17}{2}-\left(-\frac{3-\sqrt{5}}{4}\right)^{0}$$

Expand \left(-\frac{3-\sqrt{5}}{4}\right)^{0}.

$$\frac{17}{2}-\left(-1\right)^{0}\times \left(\frac{3-\sqrt{5}}{4}\right)^{0}$$

Calculate -1 to the power of 0 and get 1.

$$\frac{17}{2}-\left(\frac{3-\sqrt{5}}{4}\right)^{0}$$

To raise \frac{3-\sqrt{5}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{17}{2}-\frac{\left(3-\sqrt{5}\right)^{0}}{4^{0}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4^{0} is 2. Multiply \frac{\left(3-\sqrt{5}\right)^{0}}{4^{0}} times \frac{2}{2}.

$$\frac{17}{2}-\frac{2\left(3-\sqrt{5}\right)^{0}}{2}$$

Since \frac{17}{2} and \frac{2\left(3-\sqrt{5}\right)^{0}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{17-2\left(3-\sqrt{5}\right)^{0}}{2}$$

Calculate 3-\sqrt{5} to the power of 0 and get 1.

$$\frac{17}{2}-\frac{1}{4^{0}}$$

Calculate 4 to the power of 0 and get 1.

$$\frac{17}{2}-\frac{1}{1}$$

Anything divided by one gives itself.

$$\frac{17}{2}-1$$

Subtract 1 from \frac{17}{2} to get \frac{15}{2}.

$$\frac{15}{2}$$

Factor

$$\frac{3 \cdot 5}{2} = 7\frac{1}{2} = 7.5$$