Radicals in maths are defined with examples and detailed solutions. Questions with their solutions are also presented.\( \) \( \) Show
Power of \( n \)Let the following power of \( 2 \) (or exponent ) operation be represented by the diagram below: \( a^n = a \times a \times a .... \times a \) , n times Definition of RadicalsRadical with index 2 (or Square Root) is the inverse of power 2Let us now represent the inverse operation of the power \( 2 \) operation as shown in the diagram below. Radicals with index 3 (or Cube Root) is the inverse of power 3The inverse operation of the power \( 3 \) operation as shown in the dagram below is called the cube root. In general Radicals with index \( n \) (or \(n^{th}\) Root) is the inverse of power \( n \)We now generalize and define the radicals with index \( n \) where \( n \) is a whole number. If \( y = a^n \) , then \( \sqrt[n]{y} = a \) (I) More Examples \( y = 3^5 = 243 \) , therefore \( \sqrt[5]{243} = 3 \) \( y = 10^6 = 1000000 \) , therefore \( \sqrt[6]{1000000} = 10 \) \( y = (-2)^3 = -8\) , therefore \( \sqrt[3]{-8} = -2 \) \( y = 1^{20} = 1\) , therefore \( \sqrt[20]{1} = 1 \) in general \( \sqrt[n]{1} = 1 \) for any whole number \( n \) \( y = 0^9 = 0\) , therefore \( \sqrt[9]{0} = 0 \) in general \( \sqrt[n]{0} = 0 \) for any whole number \( n \) NOTE The relationships in (I) are not valid if \( n \) is an EVEN integer and \( y \) is a NEGATIVE number. a) \( \sqrt{-16} \) is UNDEFINED in the real numbers because no real number \( x \) exists such that \( x^2 = -16 \) since the square of a real number is always positive or zero. b) \( \sqrt[4]{-1} \) is UNDEFINED in the real numbers because no real number \( x \) exists such that \( x^4 = - 1 \) for the same reason as above. The Power and the Corresponding Radical Undo Each OtherWe say that radicals and corresponding (same index) power operations undo each other. If we apply the two operations successively, the output is equal to the input because the two operations are inverse of each other. More examples: \( (\sqrt {12})^2 = 12 \) , \( \sqrt {8^2} = 8 \) \( \sqrt[n]{a^n} = a \) , \( (\sqrt[n]{a}\;)^n = a \) (II) Questions (with solutions given below)DO NOT use the calculator to answer the follwoing questions Part 1 - Given the following:\( 2^6 = 64 \) , \( \; 3^5 = 243 \) , \( \; 5^3 = 125 \), \( \; 0^7 = 0 \), \( \; 1^{20} = 1 \), \( \; 2^9 = 512 \), \( \; 5^5 = 3125 \), \( \; 10^5 = 100000 \) , \( \; 0.1^3 = 0.001 \) find the values of the following: \( \sqrt{512} \) , \( \; \sqrt[5]{3125} \) , \( \; \sqrt[5]{243} \) , \( \; \sqrt[6]{64} \) , \( \; \sqrt[3]{0.001} \) , \( \; \sqrt[20]{1} \) , \( \; \sqrt[5]{100000} \) , \( \; \sqrt[7]{0} \) , \( \; \sqrt[3]{125} \) Part 2 - Given the following: Part 3 - Simplify the following: Solutions to the Above QuestionsPart 1Given: \( 2^6 = 64 \) , \( \; 3^5 = 243 \) , \( \; 5^3 = 125 \), \( \; 0^7 = 0 \), \( \; 1^{20} = 1 \), \( \; 2^{9} = 512 \) , \( \; 5^5 = 3125 \), \( \; 10^5 = 100000 \) , \( \; 0.1^3 = 0.001 \) \( \sqrt{512} = 9 \) because we are given \( \; 2^9 = 512 \) (remember we do not write index of radical when it is equal to \(2 \) ). \( \sqrt[5]{3125} = 5\) because \( 5^5 = 3125 \) \( \sqrt[5]{243} = 3 \) because \( 3^5 = 243 \) \( \sqrt[6]{64} = 2\) because \( 2^6 = 64 \) \( \sqrt[3]{0.001} = 0.1\) because \( 0.1^3 = 0.001 \) \( \sqrt[20]{1} = 1 \) because \( 1^{20} = 1 \) and note that for any \( n \gt 0 \) , \( \sqrt[n]{1} = 1 \) \( \sqrt[5]{100000} = 10 \) because \( 10^5 = 100000 \) \( \sqrt[7]{0} = 0 \) because \( 0^7 = 0 \) and note that for any \( n \gt 0 \) , \( \sqrt[n]{0} = 0 \) \( \sqrt[3]{125} = 3\) because \( 5^3 = 125 \) Part 2 \( 2^7 = 128 \) because we are given \( \sqrt[7]{128} = 2\) \( 0.1^7 = 0.0000001 \) because \( \sqrt[7]{0.0000001} = 0.1\) \( 6^5 = 7776 \) because \( \sqrt[5]{7776} = 6\) \( 8^2 = 64 \) because \( \sqrt{64} = 8 \) \(2^9 = 512\) because \( \sqrt[9]{512} = 2 \) \( 12^4 = 20736 \) because \( \sqrt[4]{20736} = 12 \) \(10^3 = 1000 \) because \( \sqrt[3]{1000} = 10 \) \(100^2 = 10000 \) because \( \sqrt{10000} = 100 \) Part 3 \( (\sqrt[5]{3})^5 = 3 \) because power of \( 5 \) and and radical of index \( 5 \) undo each other \(\sqrt[3]{10^3} \) because cube root (radical with index 3) and power of \( 3 \) undo each other \( (\sqrt[7]{128})^7 = 128 \) because power of \( 7 \) and and radical of index \( 7 \) undo each other What are 4 examples of radicals?Some examples of radical expressions include √2,3√72,√2x+7, and 4√1p. 2 , 72 3 , 2 x + 7 , and 1 p 4 .
What are some examples of radical equations?Part A: Solving Radical Equations. √x+8=9.. √x−4=5.. √x+7=4.. √x+3=1.. 5√x−1=0.. 3√x−2=0.. |