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The cube root of 4 is 1.5874 (up to 4 decimal places). In this article, we are going to find out its value using two different methods.
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When a number is multiplied by itself three times to give a product as 4, then that number is the cube root of 4.<\/span><\/span><\/span><\/p>\n It can be also written as \\sqrt[3]{4}<\/span> <\/span>or 4^{1 \/ 3}<\/span>.<\/span><\/span><\/p>\n After the prime factorization of 4, we get<\/span><\/span><\/p>\n 4 = 2 \u00d7 2<\/span><\/span><\/p>\n Since factor 2 is not in the group of three, 4 is not a perfect cube.<\/span><\/p>\n <\/span><\/p>\n \\sqrt[3]{4}<\/span> is an irrational number as it cannot be expressed in the form of\u00a0 \\frac{p}{q}<\/span>, \u00a0where p and q are integers.<\/span><\/p>\n <\/p>\n <\/b><\/span><\/p>\n Since 4 is not a perfect cube, we cannot use the prime factorization or estimation method to find its cube root.<\/span><\/span><\/p>\n Therefore, we have to follow the trial and error method to find the value of \\sqrt[3]{4}<\/span>.<\/span><\/span><\/p>\n There are some other methods such as Halley\u2019s methods and Newton Raphson method. But we will study about these in higher classes.<\/span><\/span><\/p>\n <\/span><\/p>\n Trial 1:\u00a0<\/strong><\/p>\n 4 lies between the perfect cubes 1 and 8.<\/span><\/p>\n Hence, the cube root of 4 lies between cube root of 1 and cube root of 8.<\/span><\/p>\n 1<\\sqrt[3]{4}<2<\/span><\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Trial 2:\u00a0<\/strong><\/p>\n We can write 4 as\u00a0 \\frac{4000}{1000}<\/span><\/span><\/p>\n <\/span><\/p>\n So, \\sqrt[3]{4}=\\frac{\\sqrt[3]{4000}}{\\sqrt[3]{1000}}=\\frac{\\sqrt[3]{4000}}{10}<\/span>.<\/span><\/p>\n <\/span><\/p>\n 4000 is also not a perfect cube and it lies between perfect cubes 3375 and 4096.<\/span><\/span><\/p>\n Hence the cube root of 4000 lies between cube root of 3375 and cube root of 4096.<\/span><\/p>\n <\/span><\/p>\n 15<\\sqrt[3]{4000}<16<\/span><\/span><\/p>\n <\/span><\/p>\n \\Rightarrow \\quad \\frac{15}{10}<\\frac{\\sqrt[3]{4000}}{10}<\\frac{16}{10}<\/span><\/span><\/p>\n <\/span><\/p>\n \\Rightarrow 1.5<\\sqrt[3]{4}<1.6<\/span><\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Trial 3:\u00a0<\/strong><\/strong><\/p>\n We can write 4 as \\frac{4000000}{1000000}<\/span>.<\/span><\/strong><\/p>\n <\/span><\/strong><\/p>\n So, \\sqrt[3]{4}=\\frac{\\sqrt[3]{4000000}}{\\sqrt[3]{1000000}}=\\frac{\\sqrt[3]{4000000}}{100}<\/span><\/span><\/strong><\/span><\/strong><\/p>\n <\/span><\/strong><\/p>\n 4000000 is not a perfect cube and it lies between the perfect cubes 3944312 and 4019679.<\/span><\/span><\/p>\n Hence the cube root of 4000000 lies between the cube root of 3944312 and the cube root of 4019679.<\/span><\/p>\n <\/span><\/p>\n 158<\\sqrt[3]{4000000}<159 <\/span><\/span><\/span><\/span><\/p>\n <\/span><\/p>\n \\Rightarrow \\quad \\frac{158}{100}<\\frac{\\sqrt[3]{4000000}}{100}<\\frac{159}{100}<\/span><\/span><\/p>\n <\/span><\/p>\n \\Rightarrow 1.58<\\sqrt[3]{4}<1.59<\/span><\/span><\/p>\n Since 4000000 is much closer to 4019679 we can assume that the approximate value of \\sqrt[3]{4}<\/span><\/span>rounded up to 2 decimal places is 1.59.<\/span><\/p>\n \\Rightarrow \\quad \\sqrt[3]{4} \\approx 1.59<\/span><\/span><\/p>\n <\/span><\/p>\n This process is time-consuming but gives values much nearer to the actual value.<\/span><\/p>\n <\/span><\/p>\n We can write 4 as \\frac{8}{2}<\/span>.<\/span><\/p>\n Here the numerator 8 is a perfect cube.<\/span><\/p>\n \\sqrt[3]{4}=\\frac{\\sqrt[3]{8}}{\\sqrt[3]{2}}=\\frac{2}{\\sqrt[3]{2}}<\/span><\/span><\/p>\n If we know that the approximate value of \\sqrt[3]{2}<\/span> is 1.26<\/span><\/p>\n Therefore, \\sqrt[3]{4}=\\frac{2}{\\sqrt[3]{2}} \\approx \\frac{2}{1.26} \\approx 1.5873<\/span><\/span><\/p>\n This value of 1.5873 is also much closer to the actual value of 1.5874. But for this method, we need to remember the value of \\sqrt[3]{2}<\/span>.<\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n 1. Find the approximate value of cube root of 108?<\/b><\/p>\n Solution<\/span><\/b><\/p>\n We know, <\/span><\/b>108 = 4 \u00d7 27<\/span><\/p>\n Taking Cube Root on both sides<\/span><\/p>\n \\sqrt[3]{108}=\\sqrt[3]{(4 \\times 27)}<\/span> \\Rightarrow \\sqrt[3]{108}=\\sqrt[3]{(4)} \\times \\sqrt[3]{(27)}<\/span><\/span><\/p>\n We already know that 3 is the cube root of 27 and 1.587 is the approximate value of cube root of 4.<\/span><\/p>\n Therefore,\\sqrt[3]{108}=1.587 \\times 3<\/span><\/p>\n \\Rightarrow \\sqrt[3]{108}=4.761<\/span><\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n [\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][et_pb_text admin_label=”Related Concepts [\/et_pb_text][et_pb_text _builder_version=”4.13.1″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\nFinding the value of \\sqrt[3]{4}<\/span><\/b><\/span><\/h2>\n
Finding the value of \\sqrt[3]{4}<\/span> when we know the value of \\sqrt[3]{2}<\/span><\/b><\/span><\/h2>\n
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Solved Examples:<\/b><\/h2>\n
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