Solve 4^2+-4^2+{left(1/5right)}^5+{left(-1/5right)}^3+{left(2/3right)}^4-{left(-2/3right)}^4-frac{9}{4}

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

Evaluate

$$\frac{371779}{12500}=29.74232$$

Solution Steps

Calculate 4 to the power of 2 and get 16.

$$16+\left(-4\right)^{2}+\left(\frac{1}{5}\right)^{5}+\left(-\frac{1}{5}\right)^{3}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Calculate -4 to the power of 2 and get 16.

$$16+16+\left(\frac{1}{5}\right)^{5}+\left(-\frac{1}{5}\right)^{3}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Add 16 and 16 to get 32.

$$32+\left(\frac{1}{5}\right)^{5}+\left(-\frac{1}{5}\right)^{3}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Calculate \frac{1}{5} to the power of 5 and get \frac{1}{3125}.

$$32+\frac{1}{3125}+\left(-\frac{1}{5}\right)^{3}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Convert 32 to fraction \frac{100000}{3125}.

$$\frac{100000}{3125}+\frac{1}{3125}+\left(-\frac{1}{5}\right)^{3}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Since \frac{100000}{3125} and \frac{1}{3125} have the same denominator, add them by adding their numerators.

$$\frac{100000+1}{3125}+\left(-\frac{1}{5}\right)^{3}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Add 100000 and 1 to get 100001.

$$\frac{100001}{3125}+\left(-\frac{1}{5}\right)^{3}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Calculate -\frac{1}{5} to the power of 3 and get -\frac{1}{125}.

$$\frac{100001}{3125}-\frac{1}{125}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Least common multiple of 3125 and 125 is 3125. Convert \frac{100001}{3125} and \frac{1}{125} to fractions with denominator 3125.

$$\frac{100001}{3125}-\frac{25}{3125}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Since \frac{100001}{3125} and \frac{25}{3125} have the same denominator, subtract them by subtracting their numerators.

$$\frac{100001-25}{3125}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Subtract 25 from 100001 to get 99976.

$$\frac{99976}{3125}+\left(\frac{2}{3}\right)^{4}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Calculate \frac{2}{3} to the power of 4 and get \frac{16}{81}.

$$\frac{99976}{3125}+\frac{16}{81}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Least common multiple of 3125 and 81 is 253125. Convert \frac{99976}{3125} and \frac{16}{81} to fractions with denominator 253125.

$$\frac{8098056}{253125}+\frac{50000}{253125}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Since \frac{8098056}{253125} and \frac{50000}{253125} have the same denominator, add them by adding their numerators.

$$\frac{8098056+50000}{253125}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Add 8098056 and 50000 to get 8148056.

$$\frac{8148056}{253125}-\left(-\frac{2}{3}\right)^{4}-\frac{9}{4}$$

Calculate -\frac{2}{3} to the power of 4 and get \frac{16}{81}.

$$\frac{8148056}{253125}-\frac{16}{81}-\frac{9}{4}$$

Least common multiple of 253125 and 81 is 253125. Convert \frac{8148056}{253125} and \frac{16}{81} to fractions with denominator 253125.

$$\frac{8148056}{253125}-\frac{50000}{253125}-\frac{9}{4}$$

Since \frac{8148056}{253125} and \frac{50000}{253125} have the same denominator, subtract them by subtracting their numerators.

$$\frac{8148056-50000}{253125}-\frac{9}{4}$$

Subtract 50000 from 8148056 to get 8098056.

$$\frac{8098056}{253125}-\frac{9}{4}$$

Reduce the fraction \frac{8098056}{253125} to lowest terms by extracting and canceling out 81.

$$\frac{99976}{3125}-\frac{9}{4}$$

Least common multiple of 3125 and 4 is 12500. Convert \frac{99976}{3125} and \frac{9}{4} to fractions with denominator 12500.

$$\frac{399904}{12500}-\frac{28125}{12500}$$

Since \frac{399904}{12500} and \frac{28125}{12500} have the same denominator, subtract them by subtracting their numerators.

$$\frac{399904-28125}{12500}$$

Subtract 28125 from 399904 to get 371779.

$$\frac{371779}{12500}$$

Factor

$$\frac{371779}{2 ^ {2} \cdot 5 ^ {5}} = 29\frac{9279}{12500} = 29.74232$$