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spin [16.1K]
3 years ago
5

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

Mathematics
1 answer:
vichka [17]3 years ago
5 0

Answer:

The expression -1+14i represents  the number 2i^4-5i^3+3i^2+\sqrt{-81} rewritten in a+bi form.

Step-by-step explanation:

The value of i is i=\sqrt{-1}[tex] or [tex]i^{2}=-1[\tex].Now [tex]i^{4} in term of i^{2}[\tex] can be written as, [tex]i^{4}=i^{2}\times i^{2}

Substituting the value,

i^{4}=\left(-1\right)\times \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, [tex]i^{3}=i^{2}\times i

Substituting the value,

i^{3}=\left(-1\right)\times i

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

\therefore i^{3}=-i

Now \sqrt{-81} can be written as follows,

\sqrt{-81}=\sqrt{\left(81\right)\times\left(-1\right)}

Applying radical multiplication rule,

\sqrt{ab}={\sqrt{a}}\sqrt{b}

\sqrt{\left(81\right)\times\left(-1\right)}={\sqrt{81}}\sqrt{-1}

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

\therefore \sqrt{\left(81\right)\times\left(-1\right)}=9i

Now substituting the above values in given expression,

2i^4-5i^3+3i^2+\sqrt{-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.

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