Video Tutorial on Multiplying Imaginary Numbers
How to Multiply Powers of I
Example 1
Simplify the following product:
$$ i^6 \cdot i^3 $$
Step 1Use the rules of exponents (in other words add 6 + 3)
$$ i^{\red{6 + 3}} = i ^9 $$
Step 2$$ i^9 \\ i ^1 \\ \boxed{i} $$
Example 2
Simplify the following product:
$$3i^5 \cdot 2i^6 $$
Step 1Group the real coefficients and the imaginary terms
$$ \blue3 \red i^5 \cdot \blue2 \red i^6 \\ ( \blue 3 \cdot \blue 2) ( \red i^5 \cdot \red i^6) $$
Step 2Multiply the real numbers and use the rules of exponents on the imaginary terms
$$ ( \blue 6 ) ( \red i^{ 5 + 6}) \\ ( \blue 6 ) ( \red i^{ 11 }) $$
Step 3$$ ( \blue 6 ) ( \red i^{ 11 }) \\ ( \blue 6 ) ( \red i^{ 3 }) \\ ( \blue 6 ) ( \red {-i}) $$
Step 4Ungroup terms
$$ \boxed{-6i} $$
How to Multiply Imaginary Numbers
Example 3
Simplify the following product:
$$ 3\sqrt{-6} \cdot 5 \sqrt{-2} $$
Step 1Group the real coefficients (3 and 5) and the imaginary terms
$$ ( \blue{ 3 \cdot 5} ) ( \red{ \sqrt{-6}} \cdot \red{ \sqrt{-2} } ) $$
Step 2Multiply the real numbers and separate out $$ \sqrt{-1}$$ also known as $$ i $$ from the imaginary numbers
$$ (\blue {15}) (\red{ \sqrt{-1}} \sqrt{6} \cdot \red{\sqrt{-1}}\sqrt{2} ) \\ (\blue {15}) (\red i \sqrt{6} \cdot \red i \sqrt{2} ) $$
Step 3Multiply real radicals and imaginary numbers
$$ (\blue {15}) (\red i \color{green}{\sqrt{6}} \cdot \red i \color{green}{ \sqrt{2} } ) \\ (15) ( \red i \cdot \red i \cdot \color{green}{\sqrt{ 12} }) \\ (15) ( \red i^2 \cdot \color{green}{\sqrt{ 12} }) $$
Step 4Simplify
$$ 15 ( \red i^2 \cdot \color{green}{\sqrt{4 } \sqrt{3} }) \\ 15 ( -1 \cdot \color{green}{2 \sqrt{3} }) \\ \boxed{ -30\sqrt{3}} $$
Look carefully at the two sample problems below
Multiplying two radicals
(link)
Sample Problem A
$ \sqrt{2} \cdot \sqrt{6} \\ \sqrt{2 \cdot 6} \\ \sqrt{12} \\ \sqrt{4} \cdot \sqrt{3} \\ \boxed{2 \sqrt{3}} $
How did we get the same result?
We got the same answer because we did something wrong in Sample Problem B
Sample Problem B
$ \sqrt{-2} \cdot \sqrt{-6} \\ \red{ \sqrt{-2 \cdot -6}} \\ \sqrt{12} \\ \sqrt{4} \cdot \sqrt{3} \\ \boxed{2 \sqrt{3}} $
You learned that you can rewrite the multiplication of radicals/square roots like $$\sqrt{2} \cdot \sqrt{6} $$ as $$\sqrt{2\cdot 6} $$
However, you can not do this with imaginary numbers (ie negative radicands ).
$ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \\ \text{ if only if }\red{a>0 \text{ and } b >0 } $
So, if the radicand is negative you cannot apply that rule.
In Sample Problem B, the radicands are negative and it is therefore incorrect to write:
$ \cancelred{\sqrt{-2} \cdot \sqrt{-6} = \sqrt{-2 \cdot -6} } $
Practice Problem Multiply Powers of I
Problem 1
Step 1
Use the rules of exponents (in other words add 4 + 11)
$$ i^4 \cdot i^{11} = i^{ \red{4 + 11} } \\ i^{15} $$
Step 2
$$ i^{ \red{15} } \\ i^{ \red{3} } \\ \red{-} i $$
More questions like Problem 1...
Problem 1.1
Step 1
$$ i^1 \cdot i^{19} = i^{ \red{1 + 19} } \\ i^{20} $$
Step 2
$$ i^{ \red{20} } \\ i^{ \red{4} } \\ \boxed{1} $$
Problem 1.2
Step 1
$$ i^{15} \cdot i^{17} = i^{ \red{15 + 17} } \\ i^{32} $$
Step 2
$$ i^{ \red{32} } \\ i^{ \red{2} } \\ \boxed{-1} $$
Problem 1.3
Step 1
$$ (i^{16})^2 = i^{\red{16 \cdot 2}} \\ i^{32} $$
Step 2
$$ i^{ \red{32} } \\ i^{ \red{2} } \\ \boxed{-1} $$
Problem 2
Step 1
This problem is like example 2.
Group the real coefficients and the imaginary terms
$$ \blue3 \red i^6 \cdot \blue 7 \red i^8 \\ (\blue 3 \cdot \blue 7)( \red i^6 \cdot \red i^8) $$
Step 2
Multiply the real numbers and use the rules of exponents to simplify the imaginary ones
$$ (\blue {21})(\red i^{ 6 + 8}) \\ (\blue {21})(\red i^{ 14 }) $$
Step 3
$$ (\blue {21})(i^{\red{ 14 }}) \\ (\blue {21})(i^{\red{ 2 }}) \\ (\blue {21})(\red{-1 }) $$
Step 4
Ungroup terms
$$ (\blue {21})(\red{-1}) \\ -21 $$
More questions like Problem 2...
Problem 2.1
Step 1
This problem is like example 2.
Group the real coefficients and the imaginary terms
$$ \blue {2} \red i^{11} \cdot \blue{10} \red i^6 \\ ( \blue 2 \cdot \blue {10})( \red i^{11} \cdot \red i^6) $$
Step 2
Multiply the real numbers and use the rules of exponents to simplify the imaginary ones
$$ ( \blue {20}) ( \red i^{ 11 + 6}) \\ ( \blue {20})( \red i^{ 17 }) $$
Step 3
$$ (\blue {20})(\red i^{ 17 }) \\ (\blue {20})(i^{\red{ 3 }}) \\ (\blue {20})(\red{-i }) $$
Step 4
Ungroup terms
$$ (\blue {20})(\red{-i}) \\ \boxed{-20i} $$
Problem 2.2
Step 1
This problem is like example 2.
Group the real coefficients and the imaginary terms
$$ (-3 i^{2})^3 \\ (\blue{-3})^3(\red{i^2})^3 $$
Step 2
Apply the the rules of exponents to imaginary and real numbers
$$ (\blue{-3})^3(\red{i^2})^3 \\ (\blue{-27})(\red{i^8}) $$
Step 3
$$ (\blue{-27})(\red{i^8}) \\ (\blue{-27})(\red{i^0}) \\ (\blue{-27})(1) $$
Step 4
Ungroup terms
$$ \boxed{ -27 } $$
Problem 3
Step 1
Group the real coefficients and the imaginary terms
$$ (\blue{4\cdot 2})(\red{\sqrt{-15}} \cdot \red{\sqrt{-3}}) $$
Step 2
Multiply the real numbers and separate out $$ \sqrt{-1}$$ also known as $$ i $$ from the imaginary numbers
$$ (\blue {8})(\red{ \sqrt{-1}} \sqrt{15} \cdot \red{\sqrt{-1}} \sqrt{3}) \\ (\blue {8})(\red{i} \sqrt{15} \cdot \red{i} \sqrt{3}) $$
Step 3
Step 4
Simplify
$$ (8)( \red i^2 \cdot \color{green}{\sqrt{ 45 } }) \\ 8 ( -1 \cdot \color{green}{\sqrt{9} \sqrt{5} }) \\ 8 ( -1 \cdot \color{green}{3 \sqrt{5} }) \\ \boxed{ -24\sqrt{5}} $$
More questions like Problem 3...
Problem 3.1
Step 1
Group the real coefficients and the imaginary terms
$$ (\blue{5} \cdot \blue{7})(\red{\sqrt{-12}}\cdot \red{\sqrt{-15}}) $$
Step 2
Multiply the real numbers and separate out $$ \sqrt{-1}$$ also known as $$ i $$ from the imaginary numbers
$$ (\blue {35}) (\red{ \sqrt{-1}} \sqrt{12} \cdot \red{\sqrt{-1}}\sqrt{15}) \\ (\blue {35}) (\red{ i} \sqrt{12} \cdot \red{{i}}\sqrt{15}) $$
Step 3
$$ (\blue{35})(\red{i} \sqrt{12} \cdot \red{{i}}\sqrt{15}) \\ 35 (\red{i^2} \cdot {\color{green}2\sqrt{3}} \cdot {\color{green}\sqrt{3} \sqrt{5}}) \\ 35 (\red{i^2} \cdot {\color{purple}2\sqrt{3}} \cdot {\color{purple}\sqrt{3} \sqrt{5}}) \\ 35 (\red{i^2} \cdot {\color{green}2} \cdot {\color{purple}3} {\color{green}\sqrt{5}}) \\ 35 (\red{i^2} \cdot 6 \color{green}{ \sqrt{5}}) $$
Step 4
Simplify
$$ 35(\red{ i^2} \cdot 6 \color{green}{\sqrt{5}}) \\ (35)(-1 \cdot 6 \color{green}{\sqrt{5}}) \\ (35)(- 6 \color{green}{\sqrt{5}}) \\ (35)(- 6 \color{green}{\sqrt{5}}) \\ \boxed{-210\sqrt{5}} $$
Problem 3.2
Step 1
Group the real coefficients and the imaginary terms
$$ (\blue{-2} \cdot \blue{7} \cdot \blue{5})(\red{\sqrt{-15}} \cdot \red{\sqrt{-3}} \cdot \red{\sqrt{-10}}) $$
Step 2
Multiply the real numbers and separate out $$ \sqrt{-1}$$ also known as $$ i $$ from the imaginary numbers
$$ (\blue {-70}) (\red{ \sqrt{-1}} \sqrt{15}\cdot \red{\sqrt{-1}}\sqrt{3} \cdot \red{\sqrt{-1}}\sqrt{10} ) \\ (\blue {-70}) (\red{i} \sqrt{15}\cdot \red{i } \sqrt{3} \cdot \red{i}\sqrt{10} ) $$
Step 3
Multiply real radicals and imaginary numbers (Note: It is often easier to simplify radicals before multiplying them. So, in this case we are doing a bit of the work that we often save for step 4)
$$ (\blue{-70})(\red{i^3} {\color{green}\sqrt{15}} \cdot {\color{green}\sqrt{3}} \cdot {\color{green}\sqrt{10}}) \\ (\blue{-70})(\red{i^3} \cdot {\color{green}\sqrt{45}} \cdot {\color{green}\sqrt{10}}) \\ (\blue{-70})(\red{i^3} \cdot {\color{purple}3\sqrt{5}} \cdot {\color{green}\sqrt{10}}) \\ \\ (\blue{-70})(\red{i^3} \cdot \color{green}{ 3\sqrt{50}} ) $$
Step 4
Simplify
$$ -70 ( \red{ i^3} \cdot 3 {\color{green}\sqrt{50}}) \\ -70 ( -i \cdot 3 {\color{green}\sqrt{50}} ) \\ -70 ( -i \cdot 3 {\color{green}\sqrt{25}\sqrt{2}}) \\ -70 ( -i \cdot 3 \cdot {\color{green}5\sqrt{2}} ) \\ -70 ( -15i \cdot {\color{green}\sqrt{2}} ) \\ \\ \boxed{ 1050i\sqrt{2}} $$
Problem 4 (challenge)
$$ \text{ Jen's Solution} \\ (3 \cdot 4)(\sqrt{-2} \cdot \sqrt{-8}) \\ ( 12 ) (\sqrt{-2 \cdot -8}) \\ \red{(12)(\sqrt{16})} \\ (12)(4) \\ 48 $$
Jen's error is highlighted in red. It is the same error that you saw above in Sample Problem B
$ \sqrt{-2} \cdot \sqrt{-8} \red{ \ne } \sqrt{-2 \cdot -8} $
Further Reading:
