From 10fbef5afc7f6fd8593f8d052668f755d20adcc8 Mon Sep 17 00:00:00 2001
From: Jonas Eckert <j.eckert@ostfalia.de>
Date: Tue, 17 Sep 2024 17:43:05 +0200
Subject: [PATCH] Tag 7

---
 tag_7.ipynb | 650 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 650 insertions(+)
 create mode 100644 tag_7.ipynb

diff --git a/tag_7.ipynb b/tag_7.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "537e03d2-319c-46c9-9b08-56e3bb7a323a",
+   "metadata": {},
+   "source": [
+    "# Tag 7\n",
+    "\n",
+    "## 16\n",
+    "\n",
+    "Erinnerung: $\\text{Basis}^\\text{Exponent}$\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$a^3*a^4$\n",
+    "\n",
+    "Gleiche Basis, die Potenzen werden addiert.\n",
+    "\n",
+    "$=a^{3+4}=a^7$\n",
+    "\n",
+    "### b)\n",
+    "\n",
+    "$a^5 +a^2$ lässt sich nicht weiter vereinfachen. (Höchstens ein $a^2$ ausklammern, aber das vereinfacht den Term nicht.)\n",
+    "\n",
+    "### c)\n",
+    "\n",
+    "$(a^3)^4$\n",
+    "\n",
+    "Bei Auswertung von links nach rechts (durch Klammern erzwungen) zweier Exponenten können diese multipliziert werden. Nicht Verwechseln mit $a^{3^4}$. Potenzen werden allgemein von rechts nach links ausgewertet, also zuerst $3^4$. $a^{3^4} = a^{(3^4)} = a^{81}$\n",
+    "\n",
+    "$ = a^3*4 = a^{12}$\n",
+    "\n",
+    "### d)\n",
+    "\n",
+    "$(b^7+b^7)=2b^7$\n",
+    "\n",
+    "### e)\n",
+    "\n",
+    "$(4a)^3$\n",
+    "\n",
+    "Ist die Basis ein Produkt kann der exponent auf jeden Factor einzelnt angewendet werden.\n",
+    "\n",
+    "$ = 4^3a^3 = 64a^3$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d5100967-2079-4f35-8fa1-3ecd2cfee541",
+   "metadata": {},
+   "source": [
+    "### f)\n",
+    "\n",
+    "$(3+a)^2 = 9 + 6a + a^2 = a^2 + 6a + 9$\n",
+    "\n",
+    "### g)\n",
+    "\n",
+    "Erinnerung $x^{-n} = \\frac{1}{x^n}$\n",
+    "\n",
+    "$\\frac{a^2*b^{-1}}{a^3*b^2} = \\frac{a^2*b^{-1}}{1} * \\frac{1}{a^3*b^2} = a^2*b^{-1}*a^{-3}*b^{-2} = a^{-1}*b^{-3} = \\frac{1}{ab^3}$\n",
+    "\n",
+    "### h)\n",
+    "\n",
+    "Erinnerung $\\sqrt[n]{x} = x^\\frac{1}{n}$\n",
+    "\n",
+    "$\\sqrt{a}a = a^{\\frac{1}{2}}*a^1= a^{\\frac{3}{2}} = \\sqrt{a^3} = \\sqrt{a}^3$\n",
+    "\n",
+    "### i)\n",
+    "\n",
+    "$(\\frac{1}{a^2})^2 = \\frac{1^2}{(a^2)^2} = \\frac{1}{a^{2*2}} = \\frac{1}{a^4} = a^{-4}$\n",
+    "\n",
+    "### j)\n",
+    "\n",
+    "$\\frac{1}{\\sqrt[3]{a}} = \\frac{1}{a^\\frac{1}{3}} = a^{-\\frac{1}{3}}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "04b53608-93dc-4ffa-8341-510834f3fff6",
+   "metadata": {},
+   "source": [
+    "### k)\n",
+    "\n",
+    "$\\frac{\\sqrt[3]{a^2}}{\\sqrt[4]{a}*\\sqrt[3]{a}*\\sqrt[12]{a}}$\n",
+    "\n",
+    "Nur der Nenner:\n",
+    "\n",
+    "$\\sqrt[4]{a}*\\sqrt[3]{a}*\\sqrt[12]{a} = a^{\\frac{1}{4}}*a^{\\frac{1}{3}}*a^{\\frac{1}{12}} = a^{\\frac{1}{4}+\\frac{1}{3}+\\frac{1}{12}}$\n",
+    "\n",
+    "Nur der Exponent:\n",
+    "\n",
+    "$\\frac{1}{4}+\\frac{1}{3}+\\frac{1}{12} = \\frac{3}{12}+\\frac{4}{12}+\\frac{1}{12} = \\frac{8}{12} = \\frac{2}{3}$\n",
+    "\n",
+    "Nun der Zähler:\n",
+    "\n",
+    "$\\sqrt[3]{a^2} = (a^2)^\\frac{1}{3} = a^{2*\\frac{1}{3}} = a^\\frac{2}{3}$\n",
+    "\n",
+    "Nenner und Zähler sind gleich. Daher sit die Lösung $1$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "11ea1402-6289-40ee-a984-2a2fb632e4d7",
+   "metadata": {},
+   "source": [
+    "### l)\n",
+    "\n",
+    "$\\sqrt{(\\frac{\\sqrt[3]{a^2}}{\\sqrt[6]{a^2}})^6}$\n",
+    "$= \\sqrt{\\frac{\\sqrt[3]{a^2}^6}{\\sqrt[6]{a^2}^6}}$\n",
+    "$= \\sqrt{\\frac{(a^2)^2}{a^2}}$\n",
+    "$= \\sqrt{\\frac{(a^2)*(a^2)}{a^2}}$\n",
+    "$= \\sqrt{a^2}$\n",
+    "$= a$\n",
+    "Der letzte Schritt gilt so, auch ohne Betragsstriche, weil die Aufgabenstellung $a > 0$ voraussetzt."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3ddc5e91-436f-4ec9-9f73-af16014c08b7",
+   "metadata": {},
+   "source": [
+    "## 17\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$\\frac{(15x^2y^{-3})^{-4}}{(25x^3y^{-6})^{-2}}$\n",
+    "$= \\frac{(3*5)^{-4}x^{-8}y^{12}}{(5^2)^{-2}x^{-6}y^{12}}$\n",
+    "$= \\frac{3^{-4}*5^{-4}x^{-2}}{5^{-4}}$\n",
+    "$= 3^{-4}x^{-2} = 81^{-1}x^{-2}=\\frac{1}{81x^2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bdb760b8-7b3f-4875-970c-f84179ff2c03",
+   "metadata": {},
+   "source": [
+    "### b)\n",
+    "\n",
+    "$\\frac{(8x^3y^{-3})^{-2}}{(12x^{-2}y^{-4})^{-3}}$\n",
+    "\n",
+    "$=\\frac{8^{-2}x^{-6}y^{6}}{12^{-3}x^6y^{12}}$\n",
+    "\n",
+    "$=\\frac{8^{-2}x^{-6}y^{6}}{12^{-2}12^{-1}x^6y^6y^6}$\n",
+    "\n",
+    "$=\\frac{2^{-2}y^{6}}{3^{-2}12^{-1}x^{6+6}y^6y^6}$\n",
+    "\n",
+    "$=\\frac{12*3^2}{2^2x^12y^6}$\n",
+    "\n",
+    "$=\\frac{3*3^2}{x^{12}y^6}$\n",
+    "\n",
+    "$=\\frac{27}{x^{12}y^6}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4bc05924-7ad8-43f2-8ff7-6b165a923663",
+   "metadata": {},
+   "source": [
+    "### c)\n",
+    "\n",
+    "$(\\sqrt[3]{a^2}-\\sqrt[3]{ab}+\\sqrt[3]{b^2})*(\\sqrt[3]{a}+\\sqrt[3]{b})$\n",
+    "\n",
+    "$= \\sqrt[3]{a^2}*\\sqrt[3]{a}-\\sqrt[3]{ab}*\\sqrt[3]{a}+\\sqrt[3]{b^2}*\\sqrt[3]{a} + \\sqrt[3]{a^2}*\\sqrt[3]{b}-\\sqrt[3]{ab}*\\sqrt[3]{b}+\\sqrt[3]{b^2}*\\sqrt[3]{b}$\n",
+    "\n",
+    "$= a-\\sqrt[3]{a^2b}+\\sqrt[3]{ab^2} + \\sqrt[3]{a^2b}-\\sqrt[3]{ab^2}+b$\n",
+    "\n",
+    "$a + b$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "03708f53-afd2-4bdb-8ab7-299e00e8cec1",
+   "metadata": {},
+   "source": [
+    "### d)\n",
+    "\n",
+    "$(\\sqrt{a+b}-\\sqrt{b})*(\\sqrt{a+b}+\\sqrt{b})$\n",
+    "\n",
+    "Dritte Binomische Formel\n",
+    "\n",
+    "$= (\\sqrt{a+b})^2-(\\sqrt{b})^2$\n",
+    "\n",
+    "$= a+b - b$\n",
+    "\n",
+    "$= a$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "21d3cd0b-b399-480e-81cf-290d350f3466",
+   "metadata": {},
+   "source": [
+    "### e)\n",
+    "\n",
+    "$(1 + \\sqrt{x} + \\sqrt{x^2} + \\sqrt{x^3})*(1-\\sqrt{x})$\n",
+    "\n",
+    "$= 1 + \\sqrt{x} + \\sqrt{x^2} + \\sqrt{x^3}$\n",
+    "\n",
+    "$-(1 + \\sqrt{x} + \\sqrt{x^2} + \\sqrt{x^3})*\\sqrt{x}$\n",
+    "\n",
+    "$= 1 + \\sqrt{x} + \\sqrt{x^2} + \\sqrt{x^3}$\n",
+    "\n",
+    "$-\\sqrt{x} - \\sqrt{x^2} - \\sqrt{x^3} - \\sqrt{x^4}$\n",
+    "\n",
+    "$= 1 - \\sqrt{x^4}$\n",
+    "\n",
+    "$= 1 - x^2$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4c85a5b2-928d-42cb-9c3e-5777ad371dd3",
+   "metadata": {},
+   "source": [
+    "## 18\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$\\frac{\\text{verfügbare Masse}}{\\text{Masse einer Sonne}} = \\frac{10^{80}}{10^{57}}= 10^{80-57}=10^{23} = \\text{einhunderttrilliarden}$\n",
+    "\n",
+    "### b)\n",
+    "\n",
+    "$\\frac{800 MW}{8 * 100 W + 1,3 kW + 400 W}$\n",
+    "\n",
+    "$= \\frac{800 *10^6W}{8 * 10^2 W + 1,3 *10^3 W + 4 * 10^2 W}$\n",
+    "\n",
+    "$= \\frac{800 *10^6W}{8 * 10^2 W + 13 *10^2 W + 4 * 10^2 W}$\n",
+    "\n",
+    "$= \\frac{800 *10^6W}{(8 + 13 + 4) * 10^2 W}$\n",
+    "\n",
+    "$= \\frac{800 *10^6W}{25 * 10^2 W}$\n",
+    "\n",
+    "$= \\frac{32 *10^6W}{10^2 W}$\n",
+    "\n",
+    "$= 32 *10^4$\n",
+    "\n",
+    "$= 320000$\n",
+    "\n",
+    "### c)\n",
+    "\n",
+    "Experiment 1\n",
+    "\n",
+    "$\\frac{14 mm}{2 ns} = \\frac{14 * 10^{-3} m}{2 * 10^{-9} s} = 7 * 10^6\\frac{m}{s}$\n",
+    "\n",
+    "$ = 7 * 10^6\\frac{m}{s} * \\frac{1km}{1000m}*\\frac{60 s}{1 min}*\\frac{60 min}{1 h}$\n",
+    "\n",
+    "$ = 7 * 10^6\\frac{m}{s} * \\frac{1km}{1000m}*\\frac{60 s}{1}*\\frac{60}{1 h}$\n",
+    "\n",
+    "$ = 7 * 10^6\\frac{m}{s} * \\frac{1}{1000}*\\frac{3600}{1}\\frac{km*s}{m*h}$\n",
+    "\n",
+    "$ = 7 * 10^6\\frac{m}{s} * \\frac{3600}{1000}\\frac{km*s}{m*h}$\n",
+    "\n",
+    "$ = 7 * 10^6\\frac{m}{s} * 3,6\\frac{km*s}{m*h}$\n",
+    "\n",
+    "$ = 7 * 10^5 * 36\\frac{km}{h}$\n",
+    "\n",
+    "$ = 252 * 10^5 \\frac{km}{h}$\n",
+    "\n",
+    "Experiment 2\n",
+    "\n",
+    "$\\frac{6 mm}{3 ps} = \\frac{6 * 10^{-3} m}{3 * 10^{-12} s} = 2 * 10^9 \\frac{m}{s}$\n",
+    "\n",
+    "$= 2 * 10^9 \\frac{m}{s} * 3,6\\frac{km*s}{m*h}$\n",
+    "\n",
+    "$= 2 * 10^8 * 36\\frac{km}{h}$\n",
+    "\n",
+    "$= 72 * 10^8 \\frac{km}{h}$\n",
+    "\n",
+    "Bei Experiment 1 ist die Geschwindigkeit geringer"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "18152741-cfce-4f79-baf6-5b1dac5d6d4c",
+   "metadata": {},
+   "source": [
+    "## 19\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$\\sqrt[3]{4}*\\sqrt[3]{2} = \\sqrt[3]{2^2*2} = \\sqrt[3]{2^3} =2$\n",
+    "\n",
+    "### b)\n",
+    "\n",
+    "$\\sqrt[6]{81} * \\sqrt[6]{9} = \\sqrt[6]{3^4*3^2} = 3$\n",
+    "\n",
+    "### c)\n",
+    "\n",
+    "$\\sqrt[13]{1,3^9} * \\sqrt[13]{1,3^4} = \\sqrt[13]{1,3^{13}} = 1,3$\n",
+    "\n",
+    "### d)\n",
+    "\n",
+    "$\\sqrt[3]{\\sqrt[2]{9^3}}$\n",
+    "\n",
+    "$ = ((9^3)^\\frac{1}{2})^\\frac{1}{3}$\n",
+    "\n",
+    "$ = (9^3)^{\\frac{1}{2}*\\frac{1}{3}}$\n",
+    "\n",
+    "$ = (9^3)^{\\frac{1}{3}*\\frac{1}{2}}$\n",
+    "\n",
+    "$ = ((9^3)^\\frac{1}{3})^\\frac{1}{2}$\n",
+    "\n",
+    "$ = 9^\\frac{1}{2}$\n",
+    "\n",
+    "$ = 3$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e8707430-3655-4581-9c8e-96f1641fd0fb",
+   "metadata": {},
+   "source": [
+    "### e)\n",
+    "\n",
+    "$\\frac{\\sqrt[10]{5120}}{\\sqrt[10]{5}}$\n",
+    "\n",
+    "$= \\sqrt[10]{\\frac{5*1024}{5}}$\n",
+    "\n",
+    "$= \\sqrt[10]{1024}$\n",
+    "\n",
+    "$= 2$\n",
+    "\n",
+    "### f)\n",
+    "\n",
+    "$\\sqrt[8]{\\sqrt[3]{4^{-8}}}$\n",
+    "\n",
+    "$ = \\sqrt[3]{\\sqrt[8]{(4^{-1})^8}}$\n",
+    "\n",
+    "$ = \\sqrt[3]{4^{-1}}$\n",
+    "\n",
+    "$ = (\\sqrt[3]{4})^{-1}$\n",
+    "\n",
+    "$ = \\frac{1}{\\sqrt[3]{4}}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3be96361-79d0-489a-87d2-9c9daa798d82",
+   "metadata": {},
+   "source": [
+    "## 20\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$\\sqrt[3]{\\sqrt[4]{x}} = (x^\\frac{1}{4})^\\frac{1}{3} = x^{\\frac{1}{4}*\\frac{1}{3}} = x^\\frac{1}{12} = \\sqrt[12]{x}$\n",
+    "\n",
+    "### b)\n",
+    "\n",
+    "$\\sqrt[3]{5}*\\sqrt[3]{\\frac{x}{5}}$\n",
+    "\n",
+    "Gleiche Potenzen (Wurzel ist auch eine Potenz), die Basen werden multipliziert.\n",
+    "\n",
+    "$= \\sqrt[3]{5*\\frac{x}{5}}$\n",
+    "\n",
+    "$= \\sqrt[3]{x}$\n",
+    "\n",
+    "### c)\n",
+    "\n",
+    "$\\sqrt[7]{\\frac{\\sqrt[3]{x^{21}a+x^{21}b}}{\\sqrt[3]{a+b}}}$\n",
+    "\n",
+    "$ = \\sqrt[7]{\\frac{\\sqrt[3]{x^{21}*(a+b)}}{\\sqrt[3]{a+b}}}$\n",
+    "\n",
+    "Regel Exponent auf ein Produkt verteilen\n",
+    "\n",
+    "$ = \\sqrt[7]{\\frac{\\sqrt[3]{x^{21}}*\\sqrt[3]{a+b}}{\\sqrt[3]{a+b}}}$\n",
+    "\n",
+    "kürzen\n",
+    "\n",
+    "$ = \\sqrt[7]{\\sqrt[3]{x^{21}}}$\n",
+    "\n",
+    "$= x$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9dbef900-f160-47db-a3b3-0dc634ed02d5",
+   "metadata": {},
+   "source": [
+    "## 21\n",
+    "\n",
+    "Es wird so erweitert, dass im Nenner die dritte Binomische Formel angewedet werden kann.\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$\\frac{a\\sqrt{b}-b\\sqrt{a}}{\\sqrt{a}-\\sqrt{b}}$\n",
+    "\n",
+    "$= \\frac{(a\\sqrt{b}-b\\sqrt{a})*(\\sqrt{a}+\\sqrt{b})}{(\\sqrt{a}-\\sqrt{b)*(\\sqrt{a}+\\sqrt{b})}}$\n",
+    "\n",
+    "$= \\frac{a\\sqrt{b}*\\sqrt{a}-b\\sqrt{a}*\\sqrt{a}+ a\\sqrt{b}*\\sqrt{b}-b\\sqrt{a}*\\sqrt{b}}{a - b}$\n",
+    "\n",
+    "$= \\frac{a\\sqrt{ab} - ab + ab - b\\sqrt{ab}}{a - b}$\n",
+    "\n",
+    "$= \\frac{a\\sqrt{ab} - b\\sqrt{ab}}{a - b}$\n",
+    "\n",
+    "$= \\frac{(a - b)\\sqrt{ab}}{a - b}$\n",
+    "\n",
+    "$= \\sqrt{ab}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5763821c-7215-4066-9aa2-7ce975770b40",
+   "metadata": {},
+   "source": [
+    "### b)\n",
+    "\n",
+    "$\\frac{2b}{\\sqrt{a+b}-\\sqrt{a-b}}$\n",
+    "\n",
+    "$= \\frac{2b*(\\sqrt{a+b}+\\sqrt{a-b})}{(a+b)-(a-b)}$\n",
+    "\n",
+    "$= \\frac{2b*(\\sqrt{a+b}+\\sqrt{a-b})}{a+b-a+b}$\n",
+    "\n",
+    "$= \\frac{2b*(\\sqrt{a+b}+\\sqrt{a-b})}{2b}$\n",
+    "\n",
+    "$= \\sqrt{a+b}+\\sqrt{a-b}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e56d1d45-f47f-4d49-8ca7-25a827a671f8",
+   "metadata": {},
+   "source": [
+    "### c)\n",
+    "\n",
+    "$\\frac{b}{a-\\sqrt{a^2-b}}$\n",
+    "\n",
+    "$=\\frac{b*(a+\\sqrt{a^2-b})}{a^2-(a^2-b)}$\n",
+    "\n",
+    "$=\\frac{b*(a+\\sqrt{a^2-b})}{a^2-a^2+b}$\n",
+    "\n",
+    "$=\\frac{b*(a+\\sqrt{a^2-b})}{b}$\n",
+    "\n",
+    "$= a+\\sqrt{a^2-b}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2d736b7a-5e4d-41cf-b922-000dca321d8a",
+   "metadata": {},
+   "source": [
+    "# 5.1\n",
+    "\n",
+    "## 1\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$x^{-n}*x^0 = x^{-n}=\\frac{1}{x^n}$\n",
+    "\n",
+    "### b)\n",
+    "\n",
+    "$x^{n+1}*x^{-(n-1)} = x^{n+1-(n-1)} = x^{n+1-n+1} = x^2$\n",
+    "\n",
+    "### c)\n",
+    "\n",
+    "$x^{-n+1}*y^{-n+1}=(xy)^{-n+1}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "33746dd2-fac7-484d-b046-4f5348c25aa1",
+   "metadata": {},
+   "source": [
+    "### d)\n",
+    "\n",
+    "$(x+y)^{n-m}*(x+y)^{n-m}=((x+y)^{n-m})^2=(x+y)^{2*(n-m)}=(x+y)^{2n-2m}$\n",
+    "\n",
+    "### e)\n",
+    "\n",
+    "$((-a)^{2n-1})^{-n-1}$\n",
+    "\n",
+    "$= (-a)^{(2n-1)*(-n-1)}$\n",
+    "\n",
+    "$= (-a)^{-2n^2+n -2n+1}$\n",
+    "\n",
+    "$= (-a)^{-2n^2-n+1}$\n",
+    "\n",
+    "### f)\n",
+    "\n",
+    "$(a^{3p})^{4p}$\n",
+    "\n",
+    "$= a^{3p*4p}$\n",
+    "\n",
+    "$= a^{12p^2}$\n",
+    "\n",
+    "### g)\n",
+    "\n",
+    "$((x-y)^{n+1})^{n+1}$\n",
+    "\n",
+    "$= (x-y)^{(n+1)*(n+1)}$\n",
+    "\n",
+    "$= (x-y)^{(n+1)^2}$\n",
+    "\n",
+    "$= (x-y)^{n^2+2n+1}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fc032af6-5690-45a8-a710-b1e1ada4c98b",
+   "metadata": {},
+   "source": [
+    "### h)\n",
+    "\n",
+    "$(3x+y)^2(3x-y)^2$\n",
+    "\n",
+    "$= ((3x+y)(3x-y))^2$\n",
+    "\n",
+    "$= (9x^2-y^2)^2$\n",
+    "\n",
+    "$= (9x^2)^2 - 2 * 9x^2 * y^2 + (y^2)^2$\n",
+    "\n",
+    "$= 81x^4 - 18x^2y^2 + y^4$\n",
+    "\n",
+    "### i)\n",
+    "\n",
+    "$\\frac{5a^9b^3}{7c^4}*\\frac{10c^3}{28a^5b^7}$\n",
+    "\n",
+    "$= \\frac{5a^4}{7c}*\\frac{5}{14b^4}$\n",
+    "\n",
+    "$= \\frac{25a^4}{98b^4c}$\n",
+    "\n",
+    "### j)\n",
+    "\n",
+    "$\\frac{(7a-7b)^5}{(a-b)}$\n",
+    "\n",
+    "$= \\frac{(7(a-b))^5}{(a-b)}$\n",
+    "\n",
+    "$= \\frac{7^5(a-b)^5}{(a-b)}$\n",
+    "\n",
+    "$= \\frac{7^5(a-b)^4*(a-b)}{(a-b)}$\n",
+    "\n",
+    "$= 7^5(a-b)^4$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "69b6a94a-6557-4362-b051-3ead119c6bf6",
+   "metadata": {},
+   "source": [
+    "## 2\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$(\\frac{\\sqrt{5}}{3})^3$\n",
+    "\n",
+    "$=\\frac{\\sqrt{5^3}}{3^3}$\n",
+    "\n",
+    "$=\\frac{\\sqrt{5^2*5}}{27}$\n",
+    "\n",
+    "$=\\frac{\\sqrt{5^2}\\sqrt{5}}{27}$\n",
+    "\n",
+    "$=\\frac{5\\sqrt{5}}{27}$\n",
+    "\n",
+    "### b)\n",
+    "\n",
+    "$(\\frac{\\sqrt{3}}{\\sqrt{5}})^4$\n",
+    "\n",
+    "$= \\frac{(\\sqrt{3})^4}{(\\sqrt{5})^4}$\n",
+    "\n",
+    "$= \\frac{3^2}{5^2}$\n",
+    "\n",
+    "$= \\frac{9}{25}$\n",
+    "\n",
+    "### c)\n",
+    "\n",
+    "$(\\frac{x\\sqrt{2}}{y\\sqrt{5}})^{-4}$\n",
+    "\n",
+    "$= (\\frac{y\\sqrt{5}}{x\\sqrt{2}})^4$\n",
+    "\n",
+    "$= \\frac{y^45^2}{x^42^2}$\n",
+    "\n",
+    "$= \\frac{25y^4}{4x^4}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5eeee264-1575-42b5-b7fe-9c33ef0f8807",
+   "metadata": {},
+   "source": [
+    "## 3\n",
+    "\n",
+    "### a)\n",
+    "\n",
+    "$\\frac{3}{\\sqrt{a}}$\n",
+    "\n",
+    "$= \\frac{3\\sqrt{a}}{a}$\n",
+    "\n",
+    "### b)\n",
+    "\n",
+    "$\\frac{a}{\\sqrt[5]{a}}$\n",
+    "\n",
+    "$= \\frac{a\\sqrt[5]{a^4}}{a}$\n",
+    "\n",
+    "$= \\sqrt[5]{a^4}$\n",
+    "\n",
+    "### c)\n",
+    "\n",
+    "$\\frac{1}{\\sqrt{a}-\\sqrt{b}}$\n",
+    "\n",
+    "$= \\frac{1*(\\sqrt{a}+\\sqrt{b})}{(\\sqrt{a}-\\sqrt{b})(\\sqrt{a}+\\sqrt{b})}$\n",
+    "\n",
+    "$= \\frac{\\sqrt{a}+\\sqrt{b}}{a - b}$\n",
+    "\n",
+    "### d)\n",
+    "\n",
+    "$\\frac{1}{1+\\sqrt{a}}$\n",
+    "\n",
+    "$= \\frac{1*(1-\\sqrt{a})}{(1+\\sqrt{a})(1-\\sqrt{a})}$\n",
+    "\n",
+    "$= \\frac{1-\\sqrt{a}}{1-a}$\n",
+    "\n",
+    "### e)\n",
+    "\n",
+    "$\\frac{5+\\sqrt{x}}{5-\\sqrt{x}}$\n",
+    "\n",
+    "$= \\frac{(5+\\sqrt{x})^2}{(5-\\sqrt{x})*(5+\\sqrt{x})}$\n",
+    "\n",
+    "$= \\frac{25 + 10\\sqrt{x} + x}{25-x}$\n",
+    "\n",
+    "### f)\n",
+    "\n",
+    "$\\frac{3 + 2\\sqrt{x}}{3 - 2\\sqrt{x}}$\n",
+    "\n",
+    "$= \\frac{(3 + 2\\sqrt{x})^2}{(3 - 2\\sqrt{x})(3 + 2\\sqrt{x})}$\n",
+    "\n",
+    "$= \\frac{9 + 12\\sqrt{x} + 4x}{9 - 4x}$"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}