What is ((1+i)^9)/((1-i)^7) ?

The solution to ((1+i)^9)/((1-i)^7) is 2

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simplify ((1+i)^9)/((1-i)^7)

simplify

\frac{( 1 + i ) ^{9}}{( 1 - i ) ^{7}}

2

Solution steps

\frac{( 1 + i ) ^{9}}{( 1 - i ) ^{7}}

Expand

(1 - i)^{7} : 8 i + 8

= \frac{( 1 + i ) ^{9}}{8 i + 8}

Factor

8 i + 8 : 8 (i + 1)

= \frac{( 1 + i ) ^{9}}{8 ( i + 1 )}

Apply the commutative law:

1 + i = i + 1

= \frac{( i + 1 ) ^{9}}{8 ( i + 1 )}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

(i + 1)^{9} = (i + 1) (i + 1)^{8}

= \frac{( i + 1 ) ( i + 1 ) ^{8}}{8 ( i + 1 )}

Cancel the common factor:

i + 1

= \frac{( i + 1 ) ^{8}}{8}

Simplify

\frac{( i + 1 ) ^{8}}{8} : 2

= 2

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