1Use the laws of indices to simplify \( 3^4 \times 3^3 \), \( x^3 \times x^5 \) and \( ab^4 \times b^3 \).
2Use the laws of indices to simplify \( (a^2 b) \times (ab^3) \), \( 3x^4 \times 2x^3 \), \( (-2x^4) \times (3x^3) \) and \( (ab) \times (a^3 b^3) \times (a^2 b) \).
3Use the laws of indices to simplify \( 4^4 \div 4^2 \), \( x^8 \div x^5 \) and \( a^4 b^4 \div a^3 b^3 \).
4Use the laws of indices to simplify \( (x^6 y^3) \div (x^3 y^3) \), \( 8x^4 \div 2x^3 \) and \( 16x^4 \div 8x^3 \).
5Use the laws of indices to simplify \( (3a)^0 \), \( (2b)^3 \), \( (-3x)^4 \) and \( (-4ab^2)^3 \).
6Use the laws of indices to simplify \( (3a^3 b^2)^2 \), \( \left( \frac{x^2}{y^2} \right)^2 \), \( \frac{(3xy)^2}{3xy} \) and \( \frac{a^{4n-2}}{a^{2(2n-1)}} \).
7Simplify \( \frac{2^2 \times 4^2}{8^2} \), \( \frac{5^3 \times 125^3}{25^3} \) and \( \frac{4^4 \times 5^5}{25^3 \times 16^2} \).
8Fill the box with the right number: \( 4^{\square} = 8^2 \), \( (3x^{\square})^2 = 9x^6 \), \( (4x)^{\square} = 1 \).
9Prove that \( \frac{x^{m+n+2} \times x^{m+n+2}}{x^{2(m+n+1)}} = x^2 \).
10Prove that \( \frac{x^{p-q+1} \times x^{q-r+1} \times x^{r-p+1}}{x^3} = 1 \) and that \( (x^{a-b})^{a+b} \times (x^{b-c})^{b+c} \times (x^{c-a})^{c+a} = 1 \).
11Find the value of \( \frac{a^m \times b^n \times c^{ab}}{m^a \times n^b \times (b^a)^c} \) when \( a = 2, b = 3, c = 1, m = 4 \) and \( n = 5 \).
12Find the value of \( (a+b+c)^{m+n} \div (m+n)^{a+b+c} \) when \( a = 2, b = 3, c = 1, m = 4 \) and \( n = 5 \).
13A children's club decides to spread a public awareness message by this rule: in one day each person tells the message to 5 more people, and the next day each of those 5 tells 5 more. Starting from one person, how many people hear the message by the fifth day? Use the laws of indices to work it out.
14Why is \( x^0 = 1 \)? Show it using the law for division.
15Write \( 3^{-4} \) as a fraction and \( \frac{1}{5^{-2}} \) as a power, then find the value of each.
16A cell divides into two every 30 minutes. Starting from one cell, how many cells are there after 3 hours?
17A student wrote \( x^3 \times x^4 = x^{12} \) and \( (x^3)^4 = x^7 \). What went wrong in each, and what are the right answers?