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Popular Algebra >

simplify (6+8i)^8

Solution

simplify

(6 + 8 i)^{8}

Solution

42197248 + 90660864 i

Solution steps

(6 + 8 i)^{8}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 6, b = 8 i

= i = 0 \sum 8 (i 8) \cdot 6^{(8 - i)} (8 i)^{i}

Expand summation

= \frac{8 !}{0 ! ( 8 - 0 ) !} \cdot 6^{8} (8 i)^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} \cdot 6^{7} (8 i)^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} \cdot 6^{6} (8 i)^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} \cdot 6^{5} (8 i)^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} \cdot 6^{4} (8 i)^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} \cdot 6^{3} (8 i)^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} \cdot 6^{2} (8 i)^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} \cdot 6^{1} (8 i)^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} \cdot 6^{0} (8 i)^{8}

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} \cdot 6^{8} (8 i)^{0} : 1679616

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} \cdot 6^{7} (8 i)^{1} : 17915904 i

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} \cdot 6^{6} (8 i)^{2} : 83607552 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} \cdot 6^{5} (8 i)^{3} : 222953472 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} \cdot 6^{4} (8 i)^{4} : 371589120 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} \cdot 6^{3} (8 i)^{5} : 396361728 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} \cdot 6^{2} (8 i)^{6} : 264241152 i^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} \cdot 6^{1} (8 i)^{7} : 100663296 i^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} \cdot 6^{0} (8 i)^{8} : 16777216 i^{8}

= 1679616 + 17915904 i + 83607552 i^{2} + 222953472 i^{3} + 371589120 i^{4} + 396361728 i^{5} + 264241152 i^{6} + 100663296 i^{7} + 16777216 i^{8}

Simplify

1679616 + 17915904 i + 83607552 i^{2} + 222953472 i^{3} + 371589120 i^{4} + 396361728 i^{5} + 264241152 i^{6} + 100663296 i^{7} + 16777216 i^{8} : 90660864 i + 42197248

= 90660864 i + 42197248

Rewrite in standard complex form:

42197248 + 90660864 i

= 42197248 + 90660864 i

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Frequently Asked Questions (FAQ)

What is (6+8i)^8 ?
The solution to (6+8i)^8 is 42197248+90660864i

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