Question

Which expression represents the number 2i4−5i3+3i2+−81‾‾‾‾√ rewritten in a+bi form?

−5+7i

5−1

−1+7i

−1+14i

Answer 1

The expression
`-1+14i` represents the number
`2i^4-5i^3+3i^2+√(-81)` rewritten in a+bi form.

Explanation:

The value of
`i` is
`i=√(-1)[tex] or [tex]i^(2)=-1[\tex].</strong></p><p>Now [tex]i^(4)` in term of
`i^(2)[\tex] can be written as, </p><p>[tex]i^(4)=i^(2)* i^(2)`

Substituting the value,

`i^(4)=\left(-1\right)* \left(-1\right)`

Product of two negative numbers is always positive.

`\therefore i^(4)=1`

Now
`i^(3)` in term of
`i^(2)[\tex] can be written as, </p><p>[tex]i^(3)=i^(2)* i`

Substituting the value,

`i^(3)=\left(-1\right)* i`

Product of one negative and one positive numbers is always negative.

`\therefore i^(3)=-i`

Now
`√(-81)` can be written as follows,

`√(-81)=√(\left(81\right)*\left(-1\right))`

Applying radical multiplication rule,

`√(ab)={√(a)}√(b)`

`√(\left(81\right)*\left(-1\right))={√(81)}√(-1)`

Now,
`\sqrt{\left(81\right)=9` and
`√(-1)}=i`

`\therefore √(\left(81\right)*\left(-1\right))=9i`

Now substituting the above values in given expression,

`2i^4-5i^3+3i^2+√(-81)=2\left(1\right)-5\left(-i\right)+3\left(-1\right)+9i`

Simplifying,

`2+5i-3+9i`

Collecting similar terms,

`2-3+5i+9i`

Combining similar terms,

`-1+14i`

The above expression is in the form of a+bi which is the required expression.

Hence, option number 4 is correct.