3.01 Complex Numbers

Practice - Multiplying and Dividing

Practice Problem #8

Complete the following problems

-3 sqrt(4x^[4] y^[3]) x 10 sqrt(7x^[3]y)

Multiply the outside coefficients and the numbers within the radicand.

__ sqrt(__ x^[_] y^[_]

Answer:

-30 sqrt(28 x^[7] y^[4]

cont.

Factor the terms under the radical sign.

=

Answer:

= -30 sqrt(4 x 7 x X^[6] x X x Y^[4]

cont.

=Remove the perfect squares.

= -30 x (__)x^[_] y^[_] sqrt(__)

Answer:

= -30 x (2)x^[3] y^[2] sqrt(7x)

cont.

Multiply the outside terms to get your final answer.

(___)x^[_]y^[_] sqrt(7x)

Answer:

Practice Problem #9

$- sqrt(12z^[4]) [ 2 sqrt(3z^{3}y) + sqrt(9zy^{3})]$

Multiply the outside coefficient (-1) by the outside coefficients of the terms in parentheses. Then, multiply the numbers within the radicands.

= __ sqrt(_z ^[_] y) - sqrt(_z^[_] y^[3])

Answer:

= -2 sqrt(36z^[7] y) - sqrt(108z^[5] y^[3])

cont.

Factor the terms under the radical signs. Identify perfect squares.

-2 sqrt(__ x z^[_] x __ x y) - sqrt(36 x __ x z^[_] x z x y^[_] x y)

Answer:

= -2 sqrt(36 x z^[6] x z x y) - sqrt(36 x 3 x z^[4] x z x y^[2] x y)

cont.

-2 sqrt(36 x Z^[6] x Z x Y) - sqrt(36 x 3 x Z^[4] x Z x Y^[2] x Y)

Remove the perfect squares

=

Answer:

-2 x 6z^[3] sqrt(yz) - 6z^[2] y sqrt(3yz)

Multiply the outside terms to simplify.

__ z^[_] sqrt(yz) - 6z^[2] y sqrt(3yz)

Answer:

-12 z^[3] sqrt(yz) - 6z^[2] y sqrt(3yz)

Practice Problem #10

Complete the following problem.

${sqrt(36x^[4]y^[7])/sqrt(3xy^[2])}$

Start by reducing the numerator and denominator.

sqrt(__

Answer:

=

=

Now, we no longer have a denominator. Factor out the perfect squares.

Remove the perfect squares to simplify.

Answer:

$2x y^[2] sqrt{3xy}$

Practice Practice #11

Reduce the numerator and the denomiator. They have a common factor of 3xy.

= $(sqrt{x^[2]}/sqrt{_ y^[_]})$

Answer:

$(sqrt{x^[2]}/sqrt{2y^[2]}$

$(sqrt{3x^[3]y}/sqrt{6xy^[3]})$ = $(sqrt{x^[2]}/sqrt{2y^[2]})$

Simplify the sqare roots in the numerator and denominator.

(__)/(__)sqrt(__)

Answer:

(x/{y}sqrt{2}

$(sqrt{3x^[3]y}/sqrt{6xy^[3]})$ = $(sqrt{x^[2]}/sqrt{2y^[2]})$ = {X/(Y)sqrt(2)}

Rationalize the denominator.

$(X/[Y]sqrt{2}) x (sqrt[_]/sqrt[_])$

Answer:

(X/[Y]sqrt{2}) x (sqrt[2]/sqrt[2])

$(sqrt{3x^[3]y}/sqrt{6xy^[3]})$ = $(sqrt{x^[2]}/sqrt{2y^[2]})$ = {X/(Y)sqrt(2)} = (X/[Y]sqrt{2}) x (sqrt[2]/sqrt[2])

Multiply and simplify.

([x]sqrt[__]/[__])

Answer:

([x]sqrt[2]/[2y])

Practice Problem #12

Complete the following problem

$(4) sqrt[5b^{8}]/(5)sqrt(12a^[3]$

Simplify the square roots in the numerator and denominator.

(4b^[_] sqrt[__]/5 x ___ sqrt[__])

Answer:

(4b^[4] sqrt[5]/5 x 2a sqrt[3a])

$(4) sqrt[5b^{8}]/(5)sqrt(12a^[3]$ = (4b^[4] sqrt[5]/5 x 2a sqrt[3a])

Rationalize the denominator.

$(4b^[4] sqrt{5}/10a sqrt{3a}) x (sqrt{__}/sqrt{__})$

Answer:

$(4b^[4] sqrt{5}/10a sqrt{3a}) x (sqrt{3a}/sqrt{3a})$

$(4) sqrt[5b^{8}]/(5)sqrt(12a^[3]$ = (4b^[4] sqrt[5]/5 x 2a sqrt[3a]) = $(4b^[4] sqrt{5}/10a sqrt{3a}) x (sqrt{3a}/sqrt{3a})$

Multiply and simplify.

(4b^[4] sqrt[__]/10a x 3a) = $(4b^[4]sqrt{__}/30 ___^[_])$ = $(2 __^[4] sqrt{15a}/___ a^[2])$

Answer:

$(4b^[4] sqrt[__]/10a x 3a) = (4b^[4]sqrt{__}/30 ___^[_]) = (2 __^[4] sqrt{15a}/___ a^[2])$