Hence, a^(4/5) is _one of the five equal factors_ (hence the fifth root) of a^4. Therefore a^(4/5) = [a^4]^(1/5). In the same way, in general, a^(p/q) = [a^p]^(1/q). Hence, _the numerator of a fractional exponent indicates the power, the denominator indicates the root_. ~To find the meaning of a zero exponent.~ Assume that Law II holds for _all_ exponents. If so, (a^m)/(a^m) = a^(m - m) = a^0. But by division, (a^m)/(a^m) = 1. Therefore a^0 = 1. Axiom I. ~To find the meaning of a negative exponent.~ Assume that Law I holds for _all_ exponents. If so, a^m x a^(-m) = a^(m - m) = a^0 = 1. Hence, a^m x a^(-m) = 1. Therefore a^(-m) = 1/(a^m). Rules: _To multiply quantities having the same base, add exponents._ _To divide quantities having the same base, subtract exponents._ _To raise a quantity to a power, multiply exponents._ _To extract a root, divide the exponent of the power by the index of the root._ 1. Find the value of 3^2 - 5 x 4^0 + 8^(-2/3) + 1^(2/5). 2. Find the value of 8^(-2/3) + 9^(3/2) - 2^(-2) + 1^(-2/5) - 7^0. Give the value of each of the following: 3. (3^0)/5, 3/(5^0), (3^0)/(5^0), 3^0 x 5, 3 x 5^0, 3^0 x 5^0, 3^0 + 5^0, 3^0 - 5^0. 4. Express 7^0 as some power of 7 divided by itself. Simplify: 5. 16^(1/3) . 2^(1/2) . 32^(5/6). (Change to the same base first.) 6. [2/(8^(-3))]^(1/5). 7. [(x^n)^(n + 2)]/[(x^(n + 1))(x^(n - 1))]. 8. (x + 3x^(2/3) - 2x^(1/3))(3 - 2x^(-1/3) + 4x^(-2/3)). 9. [(a^2b)/(c^2d)]^(1/2) x [(c^3d)/(ab^3)]^(1/3) x [(a^(1/3)c)/(b^(1/4)d^(5/12))]^2. 10. [(a^(-4))/(b^(-2)c)]^(-3/4) x [(a^(-1)b[c^(-3)]^(1/2))/(ab^(-1))]^(1/2). 11. [([a^2]^(1/3))/([b^(-1)]^(1/4)) . ([c^(-3)]^(1/2))/(a^(1/3)) . (b^(-1/4)a^(1/3))/(c^(-1))]^(-6). ~Reference:~ The chapter on Theory of Exponents in any algebra. Solve for x: 1. x^(2/3) = 4. 2. x^(-3/4) = 8. Factor: 3. x^(2/3) - 9. 4. x^(3/5) + 27. 5. x^(2a) - y^(-6). 6. a^(1/3) x^(1/2) - 3a^(1/3) + 5x^(1/2) - 15. 7. Find the H. C. F. and L. C. M. of a^2 + a^(3/2) b^(1/2) + a^(1/2) b^(3/2) - b^2, a^2 - a^(3/2) b^(1/2) - a^(1/2) b^(3/2) - b^2. 8. Simplify the product of: (ayx^(-1))^(1/2), (bxy^(-2))^(1/3), and (y^2a^(-2)b^(-2))^(1/4). (_Princeton._) 9. Find the square root of: 25a^(4/3)b^(-3) - 10a^(2/3)b^(-3/2) - 49 + 10a^(-2/3)b^(3/2) + 25a^(-4/3)b^3. 10. Simplify [(2^(n + 2))/(4^(-n)) / (8^n)/(2^3)]^(1/5). 11. Find the value of (7 . 13^0 / 7)/(21^0) + 3^0 x (4^0 . 7^0)/[(7a + b)^0] + 8^(-2/3). 12. Express as a power of 2: 8^3; 4^5; 4^3 . 8^(2/3) . 16^(3/4). 13. Simplify {[(x^(a + 1))/(x^(1 - a))]^a / [(x^a)/(x^(1 - a))]^(a - 1)}^(1/(3a - 1)). 14. Simplify [(x^(5/2) y^(4/3))/(z^(-5/4)) . (z^4)/(x^(-3) y^(-5/3)) / (y^(-2) z^(1/4))/(x^(-1/2))]^(1/5).