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# Power of Power Exponent Rule

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### Power of Power Exponent Rule

1. 1. Image Source: http://cdn.abclocal.go.com
2. 2. The World’s Population is growing at an Exponential Rate.Most of the growth is in developing countries where many peopleare having many children. How will they all be accommodated?Image Source: http://2.bp.blogspot.com
3. 3. If we have One Person and they have 4 children, and theneach of these children have 4 children, and so on, we get thefollowing “Power of Power” Exponential Population Growth.Generation 0 1 2 3 4Children 1 4 16 64 216Powers Values (20)2(21)2(22)2(23)2(24)2Rule: (2Generation)2Children 1 4 16 64 216Image Source: http://backtomyroots.wordpress.com
4. 4. The big number “2” is called the “base”and is what we multiply togetherThe little number “4” is called the“Index” or “Exponent” and tells ushow many times to multiply outthe big number “2”24= 2 x 2 x 2 x 2Multiply four of the Base Number
5. 5. Now apply the “Add Rule” for MultiplicationThe Power of “4” outside thebrackets, tells us to multiplyout what is contained in thebrackets four times.(23) 4= 23x 23x 23x 23Multiply four lots of what is in the brackets= 23 + 3 + 3 + 3= 212
6. 6. Now apply the “Add Rule” for MultiplicationThe same approach can be usedfor expanding an Algebra variableletter, with a Power of “4” outsidethe brackets.(n2) 4= n2x n2x n2x n2Multiply four lots of what is in the brackets= n2 + 2 + 2 + 2= n8
7. 7. For the brackets item (am)nIf we have a Base “a”, raised to a Power, andthen we raise this to another Power, we canskip expanding out the powers and use thefast track rule and MULTIPLY the Exponents(am)n= am x nor amnThis rule works for both letters and numbers
8. 8. The Power of PowerRule involves Multiplyingthe two Index Powers.(23)4= 23 x 4= 212(n2)4= n2 x 4= n8This rule only works if there is a single Positive Base inside the brackets.
9. 9. WARNING: The Power of Power Rule onlyworks if there is one single positive Base(eg. a number or letter) inside the brackets!(2n3)4= 2n3x4= 2n12Two Bases It is wrong to expand them like thisImages from Clker.com
10. 10. Let’s consider (22)3x (24)5We apply the Power Rule to both items:(22)3x (24)5= 22 x 3x 24 x 5= 26x 220= 220 + 6= 226We now finish our taskby using the ADD RULEfor Multiplying Powerterms which have theexact same Base.
11. 11. Simplify the expression (m3)2x (m2)5We apply the Power Rule to both items:(m3)2x (m2)5= m3 x 2x m2 x 5= m6x m10= m10 + 6= m16We now finish our taskby using the ADD RULEfor Multiplying Powerterms which have theexact same Base.
12. 12. Simplify the expression (p3)2x (q2)5We apply the Power Rule to both items:(p3)2x (q2)5= p3 x 2x q2 x 5= p6x q10= p6q10We CANNOT use theADD RULE because thePower terms do NOThave the same Base.
13. 13. Simplify using Exponent Rules (32)4(33)2We apply the Power Rule to both items:(32)4(33)2= 32 x 433 x 2= 3836= 38 - 6= 32We now finish our taskby using SUBTRACTRULE for Dividingterms which have theexact same Base.
14. 14. Simplify using Exponent Rules (h4)3(h2)2We apply the Power Rule to both items:(h4)3(h2)2= h4 x 3h2 x 2= h12h4= h12 - 4= h8We now finish our taskby using SUBTRACTRULE for Dividingterms which have theexact same Base.
15. 15. Simplify the expression (k3)2(w2)5We apply the Power Rule to both items:(k3)2(w2)5= k3 x 2w2 x 5= k6w10= k6w10We CANNOT use theSUBTRACT RULEbecause the Powerterms do NOT bothhave the same Base.