# Roots of Real Numbers and Radicals Questions with Solutions for Grade 10

Grade 10 questions on roots of numbers and radicals with solutions are presented.

 Definition x is the nth root of a number y is equivalent to xn = y. For n = 2, the nth root is called the square root. For n = 3, the nth root is called the cubic root. Examples 1) Since 32 = 9, 3 is the square (n=2) root of 9. 2) Also since (-3)2 = 9, -3 is also a square root of 9. 3) Since (-2)3 = -8, -2 is the cubic (n = 3) root of -8 4) Since 34 = 81 and (-3)4 =81, the fourth roots of $81$ are $3$ and $-3$ Properties of Roots of Real Numbers 1) For n even and y positive, there are two nth roots of y Examples Since 104=10000 and (-10)4 = 10000, the fourth roots of 10000 are 10 and -10. 2) For n even and y negative, there are no real nth roots of y. Examples The square root of - 4 is not a real number since no real number x exists such that x2 = -4 The fourth root of - 16 is not a real number since no real number x exists such that x4 = -16 3) For n odd, there is always one nth root of y. Examples The cubic (n=3) root of 8 is equal to 2. The fifth root (n=5) of -100000 is equal to -10 Principal Root For n even, the principal root is the positive root. For n odd there is only one root and it is the principal root. Examples The principal 6th of 64 is equal to 2. The principal cubic root of -64 is equal to - 4. Radical Notation The symbol √ is called a radical and is used to indicate the principal root of a number as follows: n√y where n is called the index of the radical and y is called the radicand. Examples Because of its widespread use, the square root of y is written as √y without indicating the index. Questions With Answers 18. Use a calculator to approximate the following to 3 decimal places: Solutions to the Above Problems

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