Worksheet Chapter 4: Indices, Surds and Logarithms

1. Simplify the surd expression (2√5 + 3√2)(√5 - 2√2) completely.

2. Express 4 + √32 - √3 in the form p + q√3, where p and q are integers.

3. Fill in the blank: The value of any non-zero real number raised to the power of zero, a⁰, is equal to __________.

4. True or False: According to the Laws of Logarithms, log_a (M + N) is equivalent to log_a M × log_a N.

5. Solve the logarithmic equation: log₃ (2x + 1) - log₃ (x - 1) = 2.

6. Solve the index equation: 4^(x + 1) - 9(2^x) + 2 = 0.

7. Given that log₂ 3 = m and log₂ 5 = n, express log₅ 15 in terms of m and n.

8. True or False: The radical expression √a + √b can be simplified directly to √(a + b).

9. Solve the exponential equation 3²x - 1) = 7^x correct to 4 significant figures.

10. The population P of a bacterial colony after t hours is modeled by P = 500 e⁰.4t). (a) Find the initial population of the bacteria colony. (b) Calculate the time t, in hours, taken for the population to reach 4 000 bacteria.