Index Law Practice Questions

• The following are index law practice test questions

Section A: Multiple Choice

• Any number to the power of 0 equals 1, so ( 10^0 = 1 )
• When raising a power to a power, multiply the exponents: ( (3^2)^4 = 3^{2 \times 4} = 3^8 )
• To divide powers with the same base, subtract the exponents: ( 5^7 \div 5^3 = 5^{7-3} = 5^4 )
• When multiplying powers with the same base, add the exponents: ( a^m \times a^n = a^{m+n} )
• When raising a power to a power, multiply the exponents: ( (2^5)^3 = 2^{5 \times 3} = 2^{15} )
• Any number (except 0) divided by itself equals 1: ( 6^4 \div 6^4 = 1 )
• Any number to the power of 1 equals itself: ( a^1 = a )
• When multiplying powers with the same base, add the exponents: ( 10^2 \times 10^3 = 10^{2+3} = 10^5 )
• When raising a power to a power, multiply the exponents: ( (x^3)^2 = x^{3 \times 2} = x^6 )
• A negative exponent indicates a reciprocal: ( a^{-n} = \frac{1}{a^n} ) for any nonzero number ( a )

Section B: Short Answer

• To divide powers with the same base, subtract the exponents: ( 4^6 \div 4^2 = 4^{6-2} = 4^4 )
• When raising a power to a power, multiply the exponents: ( (7^2)^3 = 7^{2 \times 3} = 7^6 )
• Any number to the power of 0 equals 1, and ( 5^2 = 25 ), so ( 8^0 + 5^2 = 1 + 25 = 26 )
• To divide powers with the same base, subtract the exponents: ( 3^5 \div 3^2 = 3^{5-2} = 3^3 )
• When multiplying powers with the same base, add the exponents: ( 9^3 \times 9^4 = 9^{3+4} = 9^7 )
• When multiplying powers with the same base, add the exponents, and when dividing, subtract the exponents: ( 2^5 \times 2^2 \div 2^4 = 2^{5+2-4} = 2^3 )
• When raising a power to a power, multiply the exponents: ( (10^3)^2 = 10^{3 \times 2} = 10^6 )
• ( (2^4)^2 = 2^{4 \times 2} = 2^8 ), and ( 2^8 \div 2^3 = 2^{8-3} = 2^5 )

Section C: Problem Solving

• The area of a rectangle is length × width: ( 6^3 \times 6^2 = 6^{3+2} = 6^5 ) square meters
• The volume of a cube is ( s^3 ), and ( s = 3^4 ), so ( V = (3^4)^3 = 3^{4 \times 3} = 3^{12} )
• After 4 hours, the population will have tripled 4 times: ( 3^5 \times 3^4 = 3^{5+4} = 3^9 )
• After 3 years, the population will have doubled 3 times: ( 2^8 \times 2^3 = 2^{8+3} = 2^{11} )
• The surface area of a cube is ( 6s^2 ), and ( s = 5^3 ), so ( 6 \times (5^3)^2 = 6 \times 5^{3 \times 2} = 6 \times 5^6 )

Section D: Extended Response

• After 5 minutes, the number of cells will be ( 2^3 \times 2^5 = 2^{3+5} = 2^8 = 256 ) cells
• Simplify the expression:
  • (\frac{(4^5 \times 4^2)}{(4^3 \times 4^2)} = \frac{4^{5+2}}{4^{3+2}} = \frac{4^7}{4^5} = 4^{7-5} = 4^2 = 16)