[{"data":1,"prerenderedAt":75},["ShallowReactive",2],{"public-page:\u002Fen\u002Falgebradan-kyskasha-anyktamalyk:en":3},{"id":4,"slug":5,"title":6,"content":7,"content_blocks":8,"media_by_id":64,"render_mode":65,"use_mathjax":66,"meta_defaults":67,"published_at":68,"updated_at":69,"article_author_name":70,"requested_lang":71,"resolved_lang":71,"available_langs":72,"localization_langs":72},"2d7b7024-2061-4bd2-81c9-cc3f08acdfc0","algebradan-kyskasha-anyktamalyk","A concise algebra reference","@post_type: theory\n@slug: algebradan-kyskasha-anyktamalyk\n@topic_code: ALGEBRA_REFERENCE\n@title: A concise algebra reference\n@summary: A concise collection of formulas for arithmetic operations, powers, roots, inequalities, absolute value, distance, and complex numbers.\n@author: QazMath\n@source_label: QazMath.net\n@source_url: https:\u002F\u002Fqazmath.net\u002Falgebradan-kyskasha-anyktamalyk\u002F\n@tags: algebra, formulas, reference\n@published_at: 2022-07-25T14:23:03+05:00\n\n@layout: 12\nc1 text: A concise algebra reference | style: heading\n---\n@layout: 6-6\nc1 text: Arithmetic operations | style: subheading\nc1 text: \"\"\" | style: paragraph\n$ab+ac=a(b+c)$\n\n$\\dfrac{\\dfrac{a}{b}}{c}=\\dfrac{a}{bc}$\n\n$\\dfrac{a}{c}+\\dfrac{c}{d}=\\dfrac{ad+bc}{bd}$\n\n$\\dfrac{a-b}{c-d}=\\dfrac{b-a}{d-c}$\n\n$\\dfrac{ab+ac}{a}=b+c, a\\neq 0$\n\n$\\dfrac{a}{\\dfrac{b}{c}}=\\dfrac{ac}{b}$\n\n$a\\left( \\dfrac{b}{c}\\right) =\\dfrac{ab}{c}$\n\n$\\dfrac{a}{b}-\\dfrac{c}{d}=\\dfrac{ad-bc}{bd}$\n\n$\\dfrac{a+b}{c}=\\dfrac{a}{c}+\\dfrac{b}{c}$\n\n$\\dfrac{\\dfrac{a}{b}}{\\dfrac{c}{d}}=\\dfrac{ad}{bc}$\n\"\"\"\nc1 text: Properties of powers | style: subheading\nc1 text: \"\"\" | style: paragraph\n$a^{n}a^{m}=a^{n+m}$\n\n$\\left( a^{n}\\right) ^{m}=a^{nm}$\n\n$\\left( ab\\right) ^{n}=a^{n}b^{n}$\n\n$a^{-n}=\\dfrac{1}{a^{n}}$\n\n$\\left( \\dfrac{a}{b}\\right) ^{-n}=\\left( \\dfrac{b}{a}\\right) ^{n}=\\dfrac{b^{n}}{a^{n}}$\n\n$\\dfrac{a^{n}}{a^{m}}=a^{n-m}=\\dfrac{1}{a^{m-n}}$\n\n$a^{0}=1,a\\neq 0$\n\n$\\left( \\dfrac{a}{b}\\right) ^{n}=\\dfrac{a^{n}}{b^{n}}$\n\n$\\dfrac{1}{a^{-n}}=a^{n}$\n\n$a ^{\\frac{n}{m}}=\\left( a ^{\\frac{1}{m}}\\right) ^{n}=\\left( a ^{n}\\right) ^{\\frac{1}{m}}$\n\"\"\"\nc1 text: Properties of roots | style: subheading\nc1 text: \"\"\" | style: paragraph\n$\\sqrt[n] {a}=a^{\\dfrac{1}{n}}$\n\n$\\sqrt[n] {\\sqrt[m] {a}}=\\sqrt[nm] {a}$\n\n$\\sqrt[n] {a^{n}}=a$ if $n$ is odd\n\n$\\sqrt[n] {a^{n}}=\\left| a\\right|$ if $n$ is even\n\n$\\sqrt[n] {ab}=\\sqrt[n] {a}\\cdot \\sqrt[n] {b}$\n\n$\\sqrt[n] {\\dfrac{a}{b}}=\\dfrac{\\sqrt[n] {a}}{\\sqrt[n] {b}}$\n\"\"\"\nc2 text: Properties of inequalities | style: subheading\nc2 text: \"\"\" | style: paragraph\nIf $a \\lt b$, then $a+c \\lt b+c$ and $a-c \\lt b-c$.\n\nIf $a \\lt b$ and $c \\gt 0$, then $ac \\lt bc$ and $\\dfrac{a}{c} \\lt\\dfrac{b}{c}$.\n\nIf $a \\lt b$ and $c \\lt 0$, then $ac \\gt bc$ and $\\dfrac{a}{c} \\gt \\dfrac{b}{c}$.\n\"\"\"\nc2 text: Properties of absolute value | style: subheading\nc2 text: \"\"\" | style: paragraph\n$\\left| a\\right| =\\begin{cases}a,a\\geq 0\\\\ -a,a \\lt 0\\end{cases}$\n\n$\\left| a\\right| \\geq 0$\n\n$\\left| ab\\right| =\\left| a\\right| \\cdot \\left| b\\right|$\n\n$\\left| -a\\right| =\\left| a\\right|$\n\n$\\left| \\dfrac{a}{b}\\right| =\\dfrac{\\left| a\\right| }{\\left| b\\right| }$\n\n$\\left| a+b\\right| \\leq \\left| a\\right| +\\left| b\\right|$ — the triangle inequality.\n\"\"\"\nc2 text: Distance formula | style: subheading\nc2 text: \"\"\" | style: paragraph\nThe distance between the points $P_{1}\\left( x_{1};y_{1}\\right)$ and $P_{2}\\left( x_{2};y_{2}\\right)$ is given by:\n\n$d\\left( P_{1},P_{2}\\right) =\\sqrt{\\left( x_{2}-x_{1}\\right) ^{2}+\\left( y_{2}-y_{1}\\right) ^{2}}$\n\"\"\"\nc2 text: Complex numbers | style: subheading\nc2 text: \"\"\" | style: paragraph\n$i=\\sqrt{-1}$, $i^{2}=-1$, $\\sqrt{-a}=i\\sqrt{a},a\\geq 0$\n\n$(a + bi) + (c + di) = a + c + (b + d) i$\n\n$(a + bi) - (c + di) = a - c + (b - d) i$\n\n$(a + bi)(c + di)=ac - bd + (ad + bc) i$\n\n$(a+bi)(a-bi)=a^{2}+b^{2}$\n\n$\\left| a+bi\\right| =\\sqrt{a^{2}+b^{2}}$ — the absolute value of a complex number.\n\n$\\left( a+bi\\right) ^{\\ast }=a-bi$ — the complex conjugate.\n\n$\\left( a+bi\\right) ^{\\ast }\\cdot \\left( a+bi\\right) =\\left| a+bi\\right| ^{2}$\n\"\"\"",[9],{"id":10,"columns":11},"row-1794115108081-8eb6f1",[12,37],{"id":13,"span":14,"blocks":15},"col-1794115108081-08b855",6,[16,21,25,28,31,34],{"id":17,"text":18,"type":19,"style":20},"block-1794115108081-8ac46b","Arithmetic operations","text","subheading",{"id":22,"text":23,"type":19,"style":24},"block-1794115108081-d2446b","$ab+ac=a(b+c)$\n\n$\\dfrac{\\frac{a}{b}}{c}=\\dfrac{a}{bc}$\n\n$\\dfrac{a}{c}+\\dfrac{c}{d}=\\dfrac{ad+bc}{bd}$\n\n$\\dfrac{a-b}{c-d}=\\dfrac{b-a}{d-c}$\n\n$\\dfrac{ab+ac}{a}=b+c, a\\neq 0$\n\n$\\dfrac{a}{\\frac{b}{c}}=\\dfrac{ac}{b}$\n\n$a\\left( \\dfrac{b}{c}\\right) =\\dfrac{ab}{c}$\n\n$\\dfrac{a}{b}-\\dfrac{c}{d}=\\dfrac{ad-bc}{bd}$\n\n$\\dfrac{a+b}{c}=\\dfrac{a}{c}+\\dfrac{b}{c}$\n\n$\\dfrac{\\frac{a}{b}}{\\frac{c}{d}}=\\dfrac{ad}{bc}$","paragraph",{"id":26,"text":27,"type":19,"style":20},"block-1794115108081-06e304","Properties of powers",{"id":29,"text":30,"type":19,"style":24},"block-1794115108081-9cc844","$a^{n}a^{m}=a^{n+m}$\n\n$\\left( a^{n}\\right) ^{m}=a^{nm}$\n\n$\\left( ab\\right) ^{n}=a^{n}b^{n}$\n\n$a^{-n}=\\dfrac{1}{a^{n}}$\n\n$\\left( \\dfrac{a}{b}\\right) ^{-n}=\\left( \\dfrac{b}{a}\\right) ^{n}=\\dfrac{b^{n}}{a^{n}}$\n\n$\\dfrac{a^{n}}{a^{m}}=a^{n-m}=\\dfrac{1}{a^{m-n}}$\n\n$a^{0}=1,a\\neq 0$\n\n$\\left( \\dfrac{a}{b}\\right) ^{n}=\\dfrac{a^{n}}{b^{n}}$\n\n$\\dfrac{1}{a^{-n}}=a^{n}$\n\n$a ^{\\frac{n}{m}}=\\left( a ^{\\frac{1}{m}}\\right) ^{n}=\\left( a ^{n}\\right) ^{\\frac{1}{m}}$",{"id":32,"text":33,"type":19,"style":20},"block-1794115108081-260b3d","Properties of roots",{"id":35,"text":36,"type":19,"style":24},"block-1794115108081-e140d6","$\\sqrt[n] {a}=a^{\\frac{1}{n}}$\n\n$\\sqrt[n] {\\sqrt[m] {a}}=\\sqrt[nm] {a}$\n\n$\\sqrt[n] {a^{n}}=a$ if $n$ is odd\n\n$\\sqrt[n] {a^{n}}=\\left| a\\right|$ if $n$ is even\n\n$\\sqrt[n] {ab}=\\sqrt[n] {a}\\cdot \\sqrt[n] {b}$\n\n$\\sqrt[n] {\\dfrac{a}{b}}=\\dfrac{\\sqrt[n] {a}}{\\sqrt[n] {b}}$",{"id":38,"span":14,"blocks":39},"col-1794115108081-e823ca",[40,43,46,49,52,55,58,61],{"id":41,"text":42,"type":19,"style":20},"block-1794115108081-dac966","Properties of inequalities",{"id":44,"text":45,"type":19,"style":24},"block-1794115108081-30aa81","If $a \\lt b$, then $a+c \\lt b+c$ and $a-c \\lt b-c$.\n\nIf $a \\lt b$ and $c \\gt 0$, then $ac \\lt bc$ and $\\dfrac{a}{c} \\lt\\dfrac{b}{c}$.\n\nIf $a \\lt b$ and $c \\lt 0$, then $ac \\gt bc$ and $\\dfrac{a}{c} \\gt \\dfrac{b}{c}$.",{"id":47,"text":48,"type":19,"style":20},"block-1794115108081-6a7f7d","Properties of absolute value",{"id":50,"text":51,"type":19,"style":24},"block-1794115108081-3ff61d","$\\left| a\\right| =\\begin{cases}a,a\\geq 0\\\\ -a,a \\lt 0\\end{cases}$\n\n$\\left| a\\right| \\geq 0$\n\n$\\left| ab\\right| =\\left| a\\right| \\cdot \\left| b\\right|$\n\n$\\left| -a\\right| =\\left| a\\right|$\n\n$\\left| \\dfrac{a}{b}\\right| =\\dfrac{\\left| a\\right| }{\\left| b\\right| }$\n\n$\\left| a+b\\right| \\leq \\left| a\\right| +\\left| b\\right|$ — the triangle inequality.",{"id":53,"text":54,"type":19,"style":20},"block-1794115108081-dcc51d","Distance formula",{"id":56,"text":57,"type":19,"style":24},"block-1794115108081-45501c","The distance between the points $P_{1}\\left( x_{1};y_{1}\\right)$ and $P_{2}\\left( x_{2};y_{2}\\right)$ is given by:\n\n$d\\left( P_{1},P_{2}\\right) =\\sqrt{\\left( x_{2}-x_{1}\\right) ^{2}+\\left( y_{2}-y_{1}\\right) ^{2}}$",{"id":59,"text":60,"type":19,"style":20},"block-1794115108081-75fbd3","Complex numbers",{"id":62,"text":63,"type":19,"style":24},"block-1794115108081-76b632","$i=\\sqrt{-1}$, $i^{2}=-1$, $\\sqrt{-a}=i\\sqrt{a},a\\geq 0$\n\n$(a + bi) + (c + di) = a + c + (b + d) i$\n\n$(a + bi) - (c + di) = a - c + (b - d) i$\n\n$(a + bi)(c + di)=ac - bd + (ad + bc) i$\n\n$(a+bi)(a-bi)=a^{2}+b^{2}$\n\n$\\left| a+bi\\right| =\\sqrt{a^{2}+b^{2}}$ — the absolute value of a complex number.\n\n$\\left( a+bi\\right) ^{\\ast }=a-bi$ — the complex conjugate.\n\n$\\left( a+bi\\right) ^{\\ast }\\cdot \\left( a+bi\\right) =\\left| a+bi\\right| ^{2}$",{},"SSG",true,{},null,"2026-11-08T05:47:32.008Z","QazMath","en",[73,74,71],"kz","ru",1794131263915]