Laws of Exponents

1
Laws of Exponents

• Let a and b be real numbers, and m and n be
  positive integers
• Product of powers
• aman amn 2322 22222 or 25
• Power of a product
• (ab)n anbn (2x)3 (2x)(2x)(2x) or 8x3
• Power of a power
• (am)n amn (22)3 (22)(22)(22) or 26

2
Laws of Exponents

• Use the laws of exponents to simplify each
  expression below
• Product of powers Now try these
• 2a53a3 6a53 6a8 (-3x2y3)(4xy2) -12x3y5
  
• Power of a product
• (3x)3 33x3 27x3 (2st3)4 24s4t34
  16s4t12
• Power of a power
• (2p3)3 23p33 8p9 (-x3)2 (-1)2x32 x6

3
Laws of Exponents

• The normal order of operations applies when
  simplifying expressions with exponents
• Exponents before multiplication
• 3u(2u2)3u5 3u23u6u5 24u12
• Distributive property
• 3t2(t3 2t2 t) (3t2)t3 (3t2)2t2 (3t2)t
  3t5 6t4 3t3

4
Laws of Exponents

• Solving equations with variables as exponents
• Apply the laws of exponents as needed and set the
  exponents on both sides of the equation equal
• 26n 22(22n)2 ? Use the power of a power law to
  simplify
• 26n 22 24n ? 26n 24n2 ? 6n 4n 2 ? n
  1
• Note that the base must be the same for this to
  work
• 22 4n 23n-1 ? Change 4 to a power of 2 (22)
• 22 22n 23n-1 ? 22n2 23n-1 ? 2n 2 3n
  1 ? n 3