Algebra  Indices and Powers  basic questions.
Test yourself 1  Solutions.
On this page, we have solutions to the expressions in the test aimed to evaluate your understanding of the five basic concepts underlying this topic.
Many of the solutions are presented showing the actual operations on the exponents. Very often we can just do these in our head  as long as they are easy to YOU  and avoid losing time. Instead of writing 1 + 3 but writing 4 instead, you save time. Any teacher will accept the 4 as being correct. If you do not write a 4, your teacher knows where you have made an error (e.g. a 3 indicates you multiplied). It is always a balance of time versus being careful. Do not overdo one of these actions at the expense of the other.
Multiplication of Powers:
Remember, by the First Law, we add the exponents attached to the same base.
1. 3a × 2b × 5c × 4d = 120 abcd  2. y^{3} × y^{5} = y^{(3+5)} = y^{8}  3. 6^{2} × 6^{5} = 6^{(25) }= 6^{3} 
4. 2k^{3} × 3k^{4} = 6k^{(3+4)} = 6k^{7}  5. 3x^{2}y × 5xy^{3}
= 15 x^{(2+1) }y^{(1+3)}=15x^{3}y^{4} 
6. 7^{2} × 7^{5} × 7^{2}
= 7^{(2+52)} = 7^{1} = 7 
7. .

8. .  9. 10x^{10} ÷ 5x^{6}
= 2x^{(106)} = 2x^{4} 
10.  11. .  12. 3j^{2} × 2k^{3}m^{2} × 3jkmn^{2}
= 18j^{(2+1) }k^{(3+1) }m^{(2+1) }n^{2} = 18j^{3}k^{4}m^{3}n^{2} 
Powers of powers:
13. (k^{3})^{5}
= k^{3×5} = k^{15} 
14. (c^{3})^{2}
= c^{(3×2)} = c^{6} 
15. (x^{½})^{2}
= x^{(½)(2)} = x^{1} = x 
16. ((d^{2})^{3})^{1}
= (d^{(2×3)})^{1} = d^{(6×1)} = d^{6} 
17. (3x^{2}y^{3}z)^{4}
= 34x^{(2×4)}y^{(3×4)}z^{4} 
18. (2m^{3})^{3} × 4m^{5}
= 2^{3}m^{(3×3)}×4m^{5} 
Negative powers:
19.  20. .  21. (3^{1})^{2} × 27
= 3^{2} × 3^{3} = 3^{1} = 3 
22.  23.  24.

Exponents of 1 or zero:
25. 5^{0} + 10^{1} = 1 + 10 = 11  26. (4m)^{0} = 1  27. 4a^{0}  7b^{0} = 4  7 = 3 
28. (3^{3})^{0} × 5^{2} = 1 × 25 = 25  29. 8m^{0} ÷ (2m)^{0} = 8 ÷ 1 = 8  30. (2x^{3 } 4y^{2})^{1} = 2x^{3}  4y^{2} 
Fractional indices:
31. .  32.  33. 
34.  35. .  36. 
Mixtures:
37. (3t)^{2} × 4t^{3 }= 9 t^{2} × 4 t^{3}
= 36 t^{5} 
38.  39. 
40. 15m^{4}n^{5} ÷ 5m^{2}n^{4}
= 3 m^{(42)} n^{(54)} = 3 m^{2} n 
41. (3g^{3})^{3} ÷ (g^{2})^{4}
=33g^{9} ÷ g^{8} = 27g 
42. (10x^{2}y^{2})^{3} ÷ (2x^{1}y^{3})^{2}
= (10^{3} x^{6} y^{6}) ÷ (2^{2} x^{2}y^{6}) = 250 x^{(6+2)} y^{(66)} = 250 x^{8 }y^{12} 