Multiply and simplify. 133) 5-6+7i 134)8i(4 3i) 135) 3i-7 + 12) 137) (4 -6)(3 -6i) Identify the conjugate of each complex number, then multiply 136) (3-4)(2 + i) 38) (1-2, 3 2 (5 3 2 3 the number and its conjugate. 139) 2 7i Divide. Write the quotient in the form a + bi. 141) 5 140) 2 3i 142) -3i 2+5i 143) 144) 9-4i 6- i 145) Simplify. 147) 0 149) 3 146) -2 +6i 148)1 150) i24<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n Let’s go through each problem step-by-step:<\/p>\n\n\n\n This expression is simply a combination of real numbers and imaginary numbers. You can combine the real terms:<\/p>\n\n\n\n [ To multiply a complex number by a real number, distribute the multiplication across the terms inside the parentheses:<\/p>\n\n\n\n [ Since (i^2 = -1), we can substitute that in:<\/p>\n\n\n\n [ Thus, the result is: Here, simplify by combining real and imaginary terms:<\/p>\n\n\n\n [ To multiply these complex numbers, apply the distributive property (FOIL):<\/p>\n\n\n\n [ Now, combine the results: We use FOIL to expand this:<\/p>\n\n\n\n [ Now, combine the terms: Use FOIL to expand:<\/p>\n\n\n\n [ Now, combine the terms: The conjugate of a complex number (a + bi) is (a – bi). So, the conjugate of (2 + 7i) is: To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator:<\/p>\n\n\n\n The conjugate of (2 + 3i) is (2 – 3i). So we multiply both the numerator and denominator by (2 – 3i):<\/p>\n\n\n\n [ First, simplify the denominator:<\/p>\n\n\n\n [ Now, simplify the numerator:<\/p>\n\n\n\n [ Thus, the result is: Multiply the numerator and denominator by the conjugate of the denominator:<\/p>\n\n\n\n The conjugate of (2 + 5i) is (2 – 5i). Multiply both the numerator and denominator by (2 – 5i):<\/p>\n\n\n\n [ Simplify the denominator:<\/p>\n\n\n\n [ Now, simplify the numerator:<\/p>\n\n\n\n [ Thus, the result is: Multiply both the numerator and denominator by the conjugate of the denominator:<\/p>\n\n\n\n The conjugate of (6 – i) is (6 + i). Multiply both the numerator and denominator by (6 + i):<\/p>\n\n\n\n [ Simplify the denominator:<\/p>\n\n\n\n [ Now, simplify the numerator using distributive property (FOIL):<\/p>\n\n\n\n [ Now, combine the terms: Thus, the result is: The process of simplifying and multiplying\/dividing complex numbers relies on applying the distributive property (FOIL) and using the fact that (i^2 = -1). For division, we multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator and express the result in the form (a + bi).<\/p>\n","protected":false},"excerpt":{"rendered":" Multiply and simplify. 133) 5-6+7i 134)8i(4 3i) 135) 3i-7 + 12) 137) (4 -6)(3 -6i) Identify the conjugate of each complex number, then multiply 136) (3-4)(2 + i) 38) (1-2, 3 2 (5 3 2 3 the number and its conjugate. 139) 2 7i Divide. Write the quotient in the form a + bi. 141) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-189241","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189241","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=189241"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189241\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=189241"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=189241"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=189241"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}133) (5 – 6 + 7i)<\/h3>\n\n\n\n
5 – 6 = -1
]
So, the result is:
[
-1 + 7i
]<\/p>\n\n\n\n134) (8i(4 + 3i))<\/h3>\n\n\n\n
8i \\times 4 = 32i
]
[
8i \\times 3i = 24i^2
]<\/p>\n\n\n\n
24i^2 = 24(-1) = -24
]<\/p>\n\n\n\n
[
32i – 24
]<\/p>\n\n\n\n135) (3i – 7 + 12)<\/h3>\n\n\n\n
-7 + 12 = 5
]
So the result is:
[
5 + 3i
]<\/p>\n\n\n\n137) ((4 – 6)(3 – 6i))<\/h3>\n\n\n\n
(4)(3) = 12
]
[
(4)(-6i) = -24i
]
[
(-6)(3) = -18
]
[
(-6)(-6i) = 36i
]<\/p>\n\n\n\n
[
12 – 24i – 18 + 36i
]
[
(12 – 18) + (-24i + 36i) = -6 + 12i
]
Thus, the result is:
[
-6 + 12i
]<\/p>\n\n\n\n136) ((3 – 4i)(2 + i))<\/h3>\n\n\n\n
(3)(2) = 6
]
[
(3)(i) = 3i
]
[
(-4i)(2) = -8i
]
[
(-4i)(i) = -4i^2 = 4 \\quad (\\text{since } i^2 = -1)
]<\/p>\n\n\n\n
[
6 + 3i – 8i + 4
]
[
(6 + 4) + (3i – 8i) = 10 – 5i
]
Thus, the result is:
[
10 – 5i
]<\/p>\n\n\n\n138) ((1 – 2i)(3 – 2i))<\/h3>\n\n\n\n
(1)(3) = 3
]
[
(1)(-2i) = -2i
]
[
(-2i)(3) = -6i
]
[
(-2i)(-2i) = 4i^2 = -4
]<\/p>\n\n\n\n
[
3 – 2i – 6i – 4
]
[
(3 – 4) + (-2i – 6i) = -1 – 8i
]
Thus, the result is:
[
-1 – 8i
]<\/p>\n\n\n\n139) (2 + 7i)<\/h3>\n\n\n\n
[
2 – 7i
]<\/p>\n\n\n\n140) (\\frac{5}{2 + 3i})<\/h3>\n\n\n\n
\\frac{5}{2 + 3i} \\times \\frac{2 – 3i}{2 – 3i} = \\frac{5(2 – 3i)}{(2 + 3i)(2 – 3i)}
]<\/p>\n\n\n\n
(2 + 3i)(2 – 3i) = 2^2 – (3i)^2 = 4 – (-9) = 4 + 9 = 13
]<\/p>\n\n\n\n
5(2 – 3i) = 10 – 15i
]<\/p>\n\n\n\n
[
\\frac{10 – 15i}{13} = \\frac{10}{13} – \\frac{15}{13}i
]<\/p>\n\n\n\n141) (\\frac{-3i}{2 + 5i})<\/h3>\n\n\n\n
\\frac{-3i}{2 + 5i} \\times \\frac{2 – 5i}{2 – 5i} = \\frac{-3i(2 – 5i)}{(2 + 5i)(2 – 5i)}
]<\/p>\n\n\n\n
(2 + 5i)(2 – 5i) = 2^2 – (5i)^2 = 4 – (-25) = 4 + 25 = 29
]<\/p>\n\n\n\n
-3i(2 – 5i) = -6i + 15i^2 = -6i – 15 = -15 – 6i
]<\/p>\n\n\n\n
[
\\frac{-15 – 6i}{29} = \\frac{-15}{29} – \\frac{6}{29}i
]<\/p>\n\n\n\n142) (\\frac{9 – 4i}{6 – i})<\/h3>\n\n\n\n
\\frac{9 – 4i}{6 – i} \\times \\frac{6 + i}{6 + i} = \\frac{(9 – 4i)(6 + i)}{(6 – i)(6 + i)}
]<\/p>\n\n\n\n
(6 – i)(6 + i) = 6^2 – i^2 = 36 – (-1) = 36 + 1 = 37
]<\/p>\n\n\n\n
(9)(6) = 54
]
[
(9)(i) = 9i
]
[
(-4i)(6) = -24i
]
[
(-4i)(i) = -4i^2 = 4
]<\/p>\n\n\n\n
[
54 + 9i – 24i + 4 = 58 – 15i
]<\/p>\n\n\n\n
[
\\frac{58 – 15i}{37} = \\frac{58}{37} – \\frac{15}{37}i
]<\/p>\n\n\n\nFinal Answers:<\/h3>\n\n\n\n
\n