(i) For n ∈ N, n even, and b > 0, there is a unique a > 0 such that an = b.
(ii) For n ∈ N, n odd, b ∈ R, there is a unique a ∈ R such that an = b. In both cases a is called the nth root of b or radical and is denoted by b1/n or n√b
(i) If n = 2, then nth root is called the square root; if n = 3, then it is called cube root.
(ii) Observe that the equation x2 = a2, has two solutions x = a, x = −a; but √a2 = |a|.
(iii) Properties of exponents given above are still valid for radicals provided each of the individual terms are defined.
(iv) Note that for n ∈ N and a ≠ 0 we have
(an)1/n = |a| if n is even, a if n is odd
(1) Simplify
(i) (125)2/3 Solution
(ii) (16)-3/4 Solution
(iii) (-1000)-2/3 Solution
(iv) (3-6)1/3 Solution
(v) 27-2/3/27-1/3 Solution
(2) Evaluate (((256)-1/2)-1/4)3 Solution
(3) Evaluate If (x1/2 + x−1/2)2 = 9/2, then find the value of (x1/2 − x−1/2) for x > 1. Solution
(4) Simplify and hence find the value of n: 32n923−n/33n = 27. Solution
(5) Find the radius of the spherical tank whose volume is 32π/3 units. Solution
(6) Simplify by rationalising the denominator.
(7 + √6) / (3 - √2) Solution
(7) Simplify
(8) Simplify
If x = √2 + √3 find (x2 + 1)/(x2 − 2) Solution
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