Exponents:Jeopardy Project by Matthew Chang

Product Rule of Powers

100

(2³)(2²)

=2²⁺³ 

=2⁵

200

(2²)(2⁷)(2⁵)

=2²⁺⁷⁺⁵ 

=2¹⁴

300

(2³5²)(2⁵5⁷)

=2³⁺⁵5²⁺⁷ 

=2⁸5⁹

400

(3⁴7³5²)(3⁶2³7⁶)

 =2³3⁴⁺⁶5²7⁶⁺³

=2³3¹⁰5²7⁹

500

(3⁵7¹³5³)(7²5¹11³)(13²3⁶11⁴)

 =3⁵⁺⁶5³⁺¹7²⁺¹³11³⁺⁴13²

=3¹¹5⁴7¹⁵11⁷13²

Quotient Rule of Powers

100

3⁵/3²

=3^5-2 

=3³

200

(3⁵)(3⁷)/3²

=3^5+7-2 

=3¹⁰

300

(2³)(3⁵)/(2⁴)(3⁷)

=2^3-43^5-7 

= 2^-13^-2

=1/(2¹)(3²)

400

(2³)(7⁵)(5³)/(2³)(7⁴)(5⁷)

=2^3-35^3-77^5-4 

=1*5^-47¹ 

=7/5⁴

500

(2⁴)(3⁷)(5⁸)(11³)/(3⁶)(7¹¹)(11⁴)(13⁶)

=(2⁴)(3^7-6)(5⁸)(11^3-3)/(3^6-6)(7¹¹)(11^4-3)(13⁶) 

=(2⁴)(3¹)(5⁸)/(7¹¹)(11¹)(13⁶)

Negative Exponents

100

5^-3

=(1/5)³ 

=1/5³

200

(3/4)^-7

=(4/3)⁷ 

=4⁷/3⁷

300

2^-35⁴7^-6

=(1/2)³(5)⁴(1/7)⁶ 

=5⁴/2³7⁶

400

(2^-4)(3⁶)/(3^-7)(2⁵)

=(1/2)⁴(3⁶)/(1/3)⁷(2⁵) 

=3¹³/2⁹

500

(2⁴/3^-2)^-3(2^-35⁷7^-11)

=(2^4*-33^-2*-3)(2^-35⁷7^-11) 

=(2^-12)(2^-35⁷7^-11)(3⁶)

=2^-12-3 5⁷ 7^-11 (3⁶)

=(1/2)¹⁵ (5⁷)(1/7)¹¹(3⁶)

=3⁶5⁷/2¹⁵7¹¹

Properties of Exponents(combo)

100

(5^-3)(5²)

=5^-3+2 

=5^-1

=(1/5)¹

=1/5

200

(3^-3)(2⁴)/(2^-2)(3⁴)

=2^4-(-2)3^-3-4 

=2⁶3^-7

=2⁶(1/3)⁷

=2⁶/3⁷

300

x³y^-4z²/(x²y³)(y²z^-2)

=x^3-2y^-4z²/x^2-2y³⁺²z^-2

=x¹y^-4+4z²⁺²/y⁵⁺⁴z^-2+2

=x¹z⁴/y⁹ 

=xz⁴/y⁹

400

(a^-3b²)(c²b³)(a²z^-4)/(x^-2y)(a^-2b^-1)(c^-3z)

=a^-3+2b²⁺³c²z^-4/x^-2ya^-2b^-1c^-3z 

=a^-1+2b⁵⁺¹c³⁺²z^-4+4/x^-2ya^-2+2b^-1+1c^-3+3z¹⁺⁴

=ab⁶c⁵/(1/x)²yz⁵

=ab⁶c⁵x²/yz⁵

500

(2³b)(c²x)(y3^-3)/(3^-5b^-1)(c^-2x^-3)(y⁷2^-2)

=(2³⁺²b¹⁺¹)(c²⁺²x¹⁺³)(y^1-1 3^-3+5)/(3^-5+5b^-1+1)(c^-2+2x^-3+3)(y^7-12^-2+2)

= (2⁵b²)(c⁴x⁴)(3²)/(y⁶)

=2⁵3²b²c⁴x⁴/y⁶

Miscellaneous Power Rules

100

(ab)²

=(a²b²) 

=a²b²

200

(2³c²)²

=2^3*2c^2*2 

=2⁶c⁴

300

(x³y²)³(x²y⁴)²

=(x^3*3y^2*3)(x^2*2y^4*2) 

= x⁹⁺⁴y⁶⁺⁸

=x¹³y¹⁴

400

(2⁰y³)²(6³y²)³(zx)³

=(2^0*2y^3*2)(6^3*3y^2*3)(z^1*3x^1*3) 

=1*x³y⁶⁺⁶z³6⁹

=x³y¹²z³6⁹

500

(3⁰x⁰)²(y³x)³(z²3⁵)³/(y²z)³(x²3⁰)⁴(x³z²)²

=(3^0*2x^0*2)(y^3*3x^1*3)(z^2*33^5*3)/(y^2*3z^1*3)(x^2*43^0*4)(x^3*2z^2*2) 

=1*1*y^9-6x^3-3z^6-63¹⁵/y^6-6z^7-6x^14-3*1

=y³3¹⁵/x¹¹z