Use x^m  x  xⁿ = x^m+n:

r³ x r = r³⁺¹ = r⁴

Use x^m/xⁿ = x^m-n :

s⁶ / s² = s^6-2 = s⁴

3.  Simplify
     (4a³)²

Use (x^m)ⁿ = x^mn and remember that 4 is squared also!

(4a³)² = 4²a^3x2 = 16a⁶

4.  Simplify
     3a³ x 2ab²

Remember to multiply the numbers first!

3a³ x 2ab² = (3 x 2) x (a³⁺¹) x b²
= 6a⁴b²

5. Simplify
    (2x³)^-3

As x^-n = 1 / xⁿ, hence this is the same as:
1 / (2x³)³ (this may help you visualise what's happening)

Using (x^m)ⁿ = x^mn, we get:
(2x³)^-3 = 2^-3x^-3x3 = 8^-1 times x^-9

this = 1 / 8 times 1 / x⁹

= 8x^-9      (or 1 / 8x⁹)

6. If x = 3⁸, express x^½ in the form 3ⁿ     where n is an integer.

7.  a = 2⁹ x 5^-6
     Express a^1/3 and a^-1 in the form
     2^m x 5ⁿ where m and n are integers.

a^1/3 = (2⁹ x 5^-6)^1/3
= 2 ^{9 x 1/3} x 5^{-6 x 1/3}
= 2³ x 5^-2

a^-1 = (2⁹ x 5^-6)^-1
= 2^{9 x -1} x 5^{-6 x -1}
= 2^-9 x 5⁶

8. Find the value of k where
    y^k = y²Öy³

y²Öy³ = y²(y³)^½ = y²(y³ x ^½)

= y² x y^3/2 = y^{2 + 3/2}

As 3/2 = 1.5, 2 + 1.5 = 3.5 = 7/2

Hence y^{2 + 3/2} = y^7/2

So, k = 7/2