Multiply (2i+3j+5k)*(i+3j+2k)

Math
(2i+3j+5k)⋅(i+3j+2k)
Expand (2i+3j+5k)(i+3j+2k) by multiplying each term in the first expression by each term in the second expression.
2ii+2i(3j)+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Simplify each term.
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Multiply 2ii.
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Raise i to the power of 1.
2(i1i)+2i(3j)+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Raise i to the power of 1.
2(i1i1)+2i(3j)+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Use the power rule aman=am+n to combine exponents.
2i1+1+2i(3j)+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Add 1 and 1.
2i2+2i(3j)+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
2i2+2i(3j)+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Rewrite i2 as -1.
2⋅-1+2i(3j)+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Multiply 2 by -1.
-2+2i(3j)+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Multiply 3 by 2.
-2+6ij+2i(2k)+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Multiply 2 by 2.
-2+6ij+4ik+3ji+3j(3j)+3j(2k)+5ki+5k(3j)+5k(2k)
Multiply j by j.
-2+6ij+4ik+3ji+3⋅3j2+3j(2k)+5ki+5k(3j)+5k(2k)
Multiply 3 by 3.
-2+6ij+4ik+3ji+9j2+3j(2k)+5ki+5k(3j)+5k(2k)
Rewrite using the commutative property of multiplication.
-2+6ij+4ik+3ji+9j2+3⋅2(jk)+5ki+5k(3j)+5k(2k)
Multiply 3 by 2.
-2+6ij+4ik+3ji+9j2+6(jk)+5ki+5k(3j)+5k(2k)
Rewrite using the commutative property of multiplication.
-2+6ij+4ik+3ji+9j2+6jk+5ki+5⋅3(kj)+5k(2k)
Multiply 5 by 3.
-2+6ij+4ik+3ji+9j2+6jk+5ki+15(kj)+5k(2k)
Multiply k by k.
-2+6ij+4ik+3ji+9j2+6jk+5ki+15kj+5⋅2k2
Multiply 5 by 2.
-2+6ij+4ik+3ji+9j2+6jk+5ki+15kj+10k2
-2+6ij+4ik+3ji+9j2+6jk+5ki+15kj+10k2
Add 6ij and 3ji.
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Move i.
-2+6ji+3ji+4ik+9j2+6jk+5ki+15kj+10k2
Add 6ji and 3ji.
-2+9ji+4ik+9j2+6jk+5ki+15kj+10k2
-2+9ji+4ik+9j2+6jk+5ki+15kj+10k2
Add 4ik and 5ki.
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Move i.
-2+9ji+9j2+6jk+4ki+5ki+15kj+10k2
Add 4ki and 5ki.
-2+9ji+9j2+6jk+9ki+15kj+10k2
-2+9ji+9j2+6jk+9ki+15kj+10k2
Add 6jk and 15kj.
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Move k.
-2+9ji+9j2+6jk+15jk+9ki+10k2
Add 6jk and 15jk.
-2+9ji+9j2+21jk+9ki+10k2
-2+9ji+9j2+21jk+9ki+10k2
Move -2.
9ji+9j2+21jk+9ki+10k2-2
Move 9ki.
9ji+9j2+21jk+10k2+9ki-2
Move 9ji.
9j2+21jk+10k2+9ji+9ki-2
Multiply (2i+3j+5k)*(i+3j+2k)

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