Algebra  Indices and Powers.
Test yourself 3  Advanced questions.
The following questions aim to help you evaluate your understanding of the five basic concepts underlying this topic. The questions are generally more difficult and require three or more steps to complete.
Use the HINTS given if you need to.
TECHNIQUE: Be systematic in your solving of these (indeed all) mathematics problems. In questions such as those below, be systemativ and move left to right by viewing the same types of terms. For example begin with collecting the numbers and then combine the first pronumeral terms together, then the second, etc.
Take your time and write out results at each step. Its better to take a few more seconds to monitor what you are doing than to rush and do too many steps at once and therefore make mistakes.
Fractions with powers:
1. Simplify:

2. Simplify and express without negative indices  3. Simplify: 
4. Solve for n: Answer.n = 0.8.  5. (i) Factorise 2^{n+1} + 2^{n}
(ii) Hence or otherwise write as a power of 2. 
6.Simplify fully: 
Powers of powers:
7. Expand and simplify:
(x^{2}y + xy^{2})^{2} 
8. Expand (1 + 3^{n})(1  3^{n})  9. Expand and simplify:
(m + m^{1})^{2} 
10. Expand and simplify:
(2g^{2}f^{3})^{2} × (2gf^{3})^{4} 
11. Expand and simplify to prove:
(3^{x})^{3} × (3^{2x})^{4} = 243^{x} 
12. If a = 3 and b = 5, compare the values for the expressions
a^{3}  2b^{2} and b^{3}  2a^{2}. 
Substitution:
Do NOT just use your calculator to obtain the answer for the following questions. If you do, you will not be practicing
the essential skills.
You will get the correct answer  but so what? What is really important? No one is watching  and you know.
The other problem is that I will haunt you with my voice coming back saying "you used it  didn't you"
13. Evaluate the following given x = 9 and y = 25: 
14. If x = 3^{2} and y = 2^{4}
express 27^{6} × 4^{8} in terms of x and y.

15. If
evaluate . 
16. What is the value of a if 
Solving equations:
17. Solve for p:
2 × 2^{p+2 }= 16 
18. Solve for x:
5^{x} × 25^{2x+1} = 125^{x} 
19. Solve for a: 
20. Solve for z:  21. Solve for x:  22. Solve for a and b: 
Fractional indices:
23. Write with a fractional index:

24. Evaluate

25.

26. Rewrite using a fractional index:

27. Write in a format with radical signs:  28. Write in a different format: 
29. (i) Expand then simplify the expression (u  v)(u^{2} + uv + v^{2}).
(ii) By letting prove the following identity: 