{"id":5174,"date":"2021-12-03T18:28:37","date_gmt":"2021-12-03T18:28:37","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=5174"},"modified":"2021-12-11T10:45:26","modified_gmt":"2021-12-11T10:45:26","slug":"a-b-3-formula-derivation-solved-examples-faq","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/a-b-3-formula-derivation-solved-examples-faq\/","title":{"rendered":"(a-b) ^3 formula \u2013 Derivation – solved examples – FAQ"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
\n” _builder_version=”4.11.3″ _module_preset=”default” header_font=”|700|||||||” header_text_align=”left” header_font_size=”50px” header_line_height=”1.18em” custom_padding=”|0px||4px|false|false” header_font_size_tablet=”” header_font_size_phone=”35px” header_font_size_last_edited=”on|phone” global_colors_info=”{}”]<\/p>\n

(a-b) ^3 formula \u2013 Derivation – solved examples – FAQ<\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n

(a-b)^3<\/span> formula<\/b><\/h2>\n

The formula for the expansion of (a-b)^3<\/span> is \\left(a^{3}+3 a b^{2}-3 a^{2} b-b^{3}\\right)<\/span>.<\/span><\/b><\/p>\n

It is the formula for the cube of the difference between two numbers.<\/span><\/p>\n

<\/span><\/p>\n

Derivation<\/b><\/h2>\n

We can write (a-b)^{3}<\/span> <\/span>as the product of (a-b)<\/span><\/span> multiplied with itself three times.<\/span><\/p>\n

(a-b)^{3}=(a-b)(a-b)(a-b)<\/span><\/span><\/p>\n

\\rightarrow(a-b)^{3}=\\left(a^{2}+b^{2}-2 a b\\right)(a-b)<\/span><\/span><\/p>\n

\\rightarrow(a-b)^{3}=\\left(a^{3}+a b^{2}-2 a^{2} b-a^{2} b-b^{3}+2 a b^{2}\\right)<\/span><\/span><\/p>\n

\\rightarrow(a-b)^{3}=\\left(a^{3}+3 a b^{2}-3 a^{2} b-b^{3}\\right)<\/span><\/span><\/p>\n

Hence the expansion of (a-b)^{3} \\text { is }\\left(a^{3}+3 a b^{2}-3 a^{2} b-b^{3}\\right)<\/span><\/span><\/p>\n

 <\/p>\n

Check<\/b><\/span><\/h2>\n

<\/b><\/span><\/p>\n

Let us take <\/span>a=9<\/span>and\u00a0 <\/span>b=5<\/span><\/p>\n

\\mathrm{LHS}=(a-b)^{3}<\/span><\/span><\/p>\n

=(9-5)^{3}<\/span><\/span><\/p>\n

=4^{3}<\/span><\/span><\/p>\n

=64<\/span><\/span><\/p>\n

<\/span><\/p>\n

\\text { RHS }=a^{3}+3 a b^{2}-3 a^{2} b-b^{3}<\/span><\/span><\/p>\n

=9^{3}+3 \\times 9 \\times 5^{2}-3 \\times 9^{2} \\times 5-5^{3}<\/span><\/span><\/p>\n

=729+675-1215-125 <\/span><\/span><\/p>\n=729+675-1215-125 <\/span>\n=64<\/span>\n

LHS = RHS<\/span><\/p>\n

\\text { Hence }(a-b)^{3}=\\left(a^{3}+3 a b^{2}-3 a^{2} b-b^{3}\\right)<\/span><\/span><\/p>\n

<\/span><\/p>\n

<\/span><\/p>\n

Physical representation<\/b><\/h2>\n

<\/b><\/p>\n

Suppose there is a cube of side \u201ca\u201d cm. If we shorten each side by \u201cb\u201d cm, then the volume of the resultant cube is equal to(a-b)^{3}<\/span><\/span><\/b><\/p>\n

<\/span><\/b><\/p>\n

Solved Examples<\/b><\/h2>\n

<\/b><\/p>\n

\\text { 1. Expand }(2 a-b)^{3}<\/span><\/b><\/p>\n

\n(2 a-b)^{3} =\\left((2 a)^{3}+3(2 a) b^{2}-3(2 a)^{2} b-b^{3}\\right) <\/span><\/b><\/p>\n

=8 a^{3}+6 a b^{2}-12 a^{2} b-b^{3}\n<\/span><\/b><\/p>\n

<\/b><\/p>\n

2. Find the value of \\left(m^{3}-n^{3}\\right)<\/span><\/span>if m-n = 5 and mn=36<\/span><\/p>\n

(m-n)^{3} =\\left(m^{3}+3 m n^{2}-3 m^{2} n-n^{3}\\right) <\/span><\/span><\/p>\n\u00a0\\rightarrow(m-n)^{3}=m^{3}-n^{3}-3 m n(m-n) <\/span>\n \\rightarrow m^{3}-n^{3}=(m-n)^{3}+3 m n(m-n) <\/span>\n

\\rightarrow m^{3}-n^{3}=(5)^{3}+3 \\times 36 \\times 5 <\/span><\/span><\/p>\n \\rightarrow m^{3}-n^{3}=125+540=665<\/span>\n

<\/span><\/p>\n

<\/span>Hence \\left(m^{3}-n^{3}\\right)<\/span><\/span>is equal to 665<\/span><\/p>\n

 <\/p>\n

3. Expand (a-3 b)^{3}+(4 a-b)^{3}<\/span><\/span><\/p>\n

(a-3 b)^{3}=\\left((a)^{3}+3(a)(3 b)^{2}-3(a)^{2}(3 b)-(3 b)^{3}\\right)<\/span><\/span><\/p>\n

=a^{3}+27 a b^{2}-9 a^{2} b-9 b^{3}<\/span><\/span><\/p>\n

(4 a-b)^{3}=\\left((4 a)^{3}+3(4 a) b^{2}-3(4 a)^{2} b-b^{3}\\right) <\/span><\/span><\/p>\n


=64 a^{3}+12 a b^{2}-48 a^{2} b-b^{3} <\/span>
<\/span><\/p>\n

(a-3 b)^{3}+(4 a-b)^{3}\u00a0 =\\left(a^{3}+27 a b^{2}-9 a^{2} b-9 b^{3}\\right)+\\left(64 a^{3}+12 a b^{2}-48 a^{2} b-b^{3}\\right) <\/span><\/span><\/p>\n


=65 a^{3}+29 a b^{2}-57 a^{2} b-10 b^{3}<\/span>
<\/span><\/span><\/p>\n

\\text { Hence, }(a-3 b)^{3}+(4 a-b)^{3} \\text { is equal to } 65 a^{3}+29 a b^{2}-57 a^{2} b-10 b^{3}<\/span><\/span><\/p>\n

 <\/p>\n

<\/span><\/p>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    Q1: What is the expansion for (a-b)^{3}<\/span>?
    <\/strong><\/h3>\n

    Ans: <\/strong>The expansion of (a-b)^{3} \\text { is }\\left(a^{3}+3 a b^{2}-3 a^{2} b-b^{3}\\right)<\/span><\/span><\/span><\/p>\n

    Q2. What is the expansion for (a-b)^{2}<\/span>?
    <\/strong><\/h3>\n

    Ans: <\/strong>The expansion of (a-b)^{2} \\text { is }\\left(a^{2}-2 a b-b^{2}\\right)<\/span><\/strong><\/p>\n

    <\/strong><\/p>\n

    Q3. What is the expansion for a^{2}-b^{2}<\/span>?<\/strong><\/h3>\n

    Ans: The expansion of a^{2}-b^{2} \\text { is }(a-b)(a+b)<\/span><\/span><\/strong><\/p>\n

    [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

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