Solve {l}{-(3/7)^1+(-3/7)^2}{+(-3/7)^3+(-3/7)^4}

$$\left. \begin{array} { l } { - ( \frac { 3 } { 7 } ) ^ { 1 } + ( - \frac { 3 } { 7 } ) ^ { 2 } } \\ { + ( - \frac { 3 } { 7 } ) ^ { 3 } + ( - \frac { 3 } { 7 } ) ^ { 4 } } \end{array} \right.$$

Evaluate

$$-\frac{696}{2401}\approx -0.289879217$$

Solution Steps

Calculate \frac{3}{7} to the power of 1 and get \frac{3}{7}.

$$-\frac{3}{7}+\left(-\frac{3}{7}\right)^{2}+\left(-\frac{3}{7}\right)^{3}+\left(-\frac{3}{7}\right)^{4}$$

Calculate -\frac{3}{7} to the power of 2 and get \frac{9}{49}.

$$-\frac{3}{7}+\frac{9}{49}+\left(-\frac{3}{7}\right)^{3}+\left(-\frac{3}{7}\right)^{4}$$

Least common multiple of 7 and 49 is 49. Convert -\frac{3}{7} and \frac{9}{49} to fractions with denominator 49.

$$-\frac{21}{49}+\frac{9}{49}+\left(-\frac{3}{7}\right)^{3}+\left(-\frac{3}{7}\right)^{4}$$

Since -\frac{21}{49} and \frac{9}{49} have the same denominator, add them by adding their numerators.

$$\frac{-21+9}{49}+\left(-\frac{3}{7}\right)^{3}+\left(-\frac{3}{7}\right)^{4}$$

Add -21 and 9 to get -12.

$$-\frac{12}{49}+\left(-\frac{3}{7}\right)^{3}+\left(-\frac{3}{7}\right)^{4}$$

Calculate -\frac{3}{7} to the power of 3 and get -\frac{27}{343}.

$$-\frac{12}{49}-\frac{27}{343}+\left(-\frac{3}{7}\right)^{4}$$

Least common multiple of 49 and 343 is 343. Convert -\frac{12}{49} and \frac{27}{343} to fractions with denominator 343.

$$-\frac{84}{343}-\frac{27}{343}+\left(-\frac{3}{7}\right)^{4}$$

Since -\frac{84}{343} and \frac{27}{343} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-84-27}{343}+\left(-\frac{3}{7}\right)^{4}$$

Subtract 27 from -84 to get -111.

$$-\frac{111}{343}+\left(-\frac{3}{7}\right)^{4}$$

Calculate -\frac{3}{7} to the power of 4 and get \frac{81}{2401}.

$$-\frac{111}{343}+\frac{81}{2401}$$

Least common multiple of 343 and 2401 is 2401. Convert -\frac{111}{343} and \frac{81}{2401} to fractions with denominator 2401.

$$-\frac{777}{2401}+\frac{81}{2401}$$

Since -\frac{777}{2401} and \frac{81}{2401} have the same denominator, add them by adding their numerators.

$$\frac{-777+81}{2401}$$

Add -777 and 81 to get -696.

$$-\frac{696}{2401}$$

Factor

$$-\frac{696}{2401} = -0.28987921699291963$$