Exercises 6 (Indices)

  1. Using the rules of indices put the following expressions in their simplest form:

    1. \(\dfrac{9^2\times 9\times 9^5}{3^9\times 3}\)
    2. \(\dfrac{7^6\times 7^3\times 4^8}{7^8\times 4^2\times 4^3}\)

  2. Simplify the following expressions without using a calculator:

    1. \(\dfrac{(7.5\times 10^2)(1.2\times 10^3)}{4}\)
    2. \(\dfrac{(4.5\times 10^{-7})(1.2\times 10^9)}{9}\)
    3. \(\dfrac{(3.3\times10^{-5})(4.2\times 10^6)}{(1.1\times10^{-2})(2.1\times 10^3)}\)
    4. \(\dfrac{4^{\frac12}\times 64^{\frac23}\times 32^{\frac15}}{16^{\frac32}\times 81^{-\frac34}}\)

  3. Simplify the following:

    1. \((a^2 x^3 y^{-2})^3 \times (a^{-3}xy^3)^{1/2}\div (axy^{-3})^{5/2}\)
    2. \(\dfrac{\sqrt[3]{a}b^{-\frac12}}{\sqrt{a^5}\sqrt{b}}\div\dfrac{ab^2c^{-\frac32}}{\sqrt{a^2b^3}c^{\frac52}}\)

  4. The e.m.f. induced in a circuit when \(N\) lines of induction are cut in a time \(t\) seconds is given by: \[e.m.f. = \frac{N}{t\times 10^8}.\] Determine the e.m.f. induced when \(N=36\times 10^8\) and \(t=30\,\mathrm{ms}\).

  5. Without using a calculator, determine the value of \(\left(\dfrac{81^{\frac14}\times 9^{\frac12}}{3^2\times 27^2}\right)^{-1}\).

  6. Simplify:

    1. \(5^7\times 5^{13}\)
    2. \(9^8\times 9^5\)
    3. \(11^2\times11^3\times11^4\)
    4. \(\dfrac{15^3}{15^2}\)
    5. \(\dfrac{4^{18}}{4^9}\)
    6. \(\dfrac{5^{20}}{5^{19}}\)
    7. \(a^7a^3\)
    8. \(a^4a^5\)
    9. \(b^{11}b^{10}b\)
    10. \(x^{7}\times x^{8}\)
    11. \(y^4\times y^8\times y^9\)

  7. Simplify:

    1. \((7^3)^2\)
    2. \((4^2)^8\)
    3. \((7^9)^2\)
    4. \(\dfrac{1}{(5^3)^8}\)
    5. \((x^2y^3)(x^3y^2)\)
    6. \((a^2bc^2)(b^2ca)\)

  8. Remove the brackets from:

    1. \((x^2y^4)^5\)
    2. \((9x^3)^2\)
    3. \((-3x)^3\)
    4. \((-x^2y^3)^4\)

  9. Simplify:

    1. \(\dfrac{(z^2)^3}{z^3}\)
    2. \(\dfrac{(y^3)^2}{(y^2)^2}\)
    3. \(\dfrac{(x^3)^2}{(x^2)^3}\)

  10. Write each of the following using only a positive power:

    1. \(x^{-4}\)
    2. \(\dfrac{1}{x^{-5}}\)
    3. \(x^{-7}\)
    4. \(y^{-2}\)
    5. \(\dfrac{1}{y^{-1}}\)

  11. Simplify the following and write your results using only positive powers:

    1. \(x^{-1}x^{-2}\)
    2. \(x^{-3}x^{-2}\)
    3. \(x^3x^{-4}\)
    4. \(x^{-4}x^9\)
    5. \(\dfrac{x^{-2}}{x^{11}}\)
    6. \((x^{-4})^2\)
    7. \((x^{-3})^3\)
    8. \((x^2)^{-2}\)

  12. Simplify:

    1. \(a^{13}a^{-2}\)
    2. \(x^{-9}x^{-7}\)
    3. \(x^{-21}x^2x\)

  13. Evaluate:

    1. \(10^{-3}\)
    2. \(10^{-4}\)
    3. \(10^{-5}\)
    4. \(\dfrac{4^{-8}}{4^{-6}}\)
    5. \(\dfrac{3^{-5}}{3^{-8}}\)

  14. Simplify, then evaluate:

    1. \(64^{\frac13}\)
    2. \(144^{\frac12}\)
    3. \(16^{-\frac14}\)
    4. \(25^{-\frac12}\)
    5. \(\left(3^{-\frac12}\right)^4\)
    6. \(\left(8^{\frac13}\right)^{-1}\)

Solutions: 1. (i) \(3^6\); (ii) \(7\times 4^3\); 2. (i) \(2.25\times 10^5\); (ii) \(60\); (iii) \(6\); (iv) \(27\); 3. (i) \(a^2x^7y^3\); (ii) \(\frac{c^4}{a^{13/6}b^{3/2}}\); 4. \(1.2\times 10^3\); 5. \(729\); 6. (a) \(5^{20}\); (b) \(9^{13}\); (c) \(11^9\); (d) \(15\); (e) \(4^9\); (f) \(5\); (g) \(a^{10}\); (h) \(a^9\); (i) \(b^{22}\); (j) \(x^{15}\); (k) \(y^{21}\) 7. (a) \(7^6\); (b) \(4^{16}\); (c) \(7^{18}\); (d) \(5^{-24}\); (e) \(x^5y^5\); (f) \(a^3b^3c^3=(abc)^3\); 8. (a) \(x^{10}y^{20}\); (b) \(81x^6\); (c) \(-27x^3\); (d) \(x^8y^{12}\); 9. (a) \(z^3\); (b) \(y^2\); (c) \(1\); 10. (a) \(\frac{1}{x^4}\); (b) \(x^5\); (c) \(\frac{1}{x^7}\); (d) \(\frac{1}{y^2}\); (e) \(y\); 11. (a) \(\frac{1}{x^3}\); (b) \(\frac{1}{x^5}\); (c) \(\frac{1}{x}\); (d) \(x^5\); (e) \(\frac{1}{x^{13}}\); (f) \(\frac{1}{x^8}\); (g) \(\frac{1}{x^9}\); (h) \(\frac{1}{x^4}\); 12. (a) \(a^{11}\); (b) \(\frac{1}{x^{16}}\); (c) \(\frac{1}{x^{18}}\); 13. (a) \(0.001\); (b) \(0.0001\); (c) \(0.00001\); (d) \(1/16\); (e) \(27\); 14. (a) \(4\); (b) \(12\); (c) \(1/2\); (d) \(1/5\); (e) \(1/9\); (f) \(1/2\)