Exponents and Powers

Exponents give us a short way to write repeated multiplication of the same number. In 2^5, the number 2 is the base and 5 is the exponent, so 2^5 means 2 x 2 x 2 x 2 x 2 = 32 and is read as 2 raised to the power 5. Large numbers, such as the mass of the Earth or the distance from the Earth to the Sun, become easy to read and compare when written using powers of 10. A negative base raised to an even power gives a positive number, while an odd power keeps the negative sign, so (-1)^4 = 1 and (-1)^3 = -1. The chapter develops the laws of exponents for non-zero bases: a^m x a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn), a^m x b^m = (ab)^m, a^m / b^m = (a/b)^m and a^0 = 1. You will use them to simplify expressions, write numbers in expanded form with powers of 10, and express large numbers in standard form, such as 150,000,000,000 m = 1.5 x 10^11 m.

Detailed Worksheet: Exponents and Powers Section A - Definitions (10 marks) 1. In 7^4, identify the base and the exponent. Write it in expanded form and read it aloud. (2 marks) 2. State the laws of exponents for multiplying and dividing powers with the same base. (2 marks) 3. What is the value of any non-zero number raised to the power 0? Explain using a^3 / a^3. (2 marks) 4. What is the standard form of a number? Write 5,985 in standard form. (2 marks) 5. What is the sign of (-a)^n when n is even and when n is odd? Give an example of each. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the value of (i) 2^6 (ii) 9^3 (iii) 11^2 (iv) (-3)^4 (v) (-2)^5 (vi) 5^4. (3 marks) 7. Express in exponential form: (i) 512 as a power of 2 (ii) 343 as a power of 7 (iii) 729 as a power of 3. (3 marks) 8. Simplify using laws of exponents: (i) 3^2 x 3^4 x 3^8 (ii) 6^15 / 6^10 (iii) (2^3)^4 (iv) a^3 x a^2 (v) 7^x x 7^2 (vi) (5^2)^3 / 5^3. (3 marks) 9. Express in standard form: (i) 3,43,000 (ii) 70,40,00,000 (iii) the diameter of the Earth, 12,756,000 m (iv) the distance of the Moon from the Earth, 384,000,000 m. (3 marks) 10. Write in expanded form using powers of 10: (i) 279404 (ii) 3006194 (iii) 20068. Also find the number for 8 x 10^4 + 6 x 10^3 + 0 x 10^2 + 4 x 10^1 + 5 x 10^0. (3 marks) Section C - Diagrams (10 marks) 11. Draw a factor tree for 1296 and use it to express 1296 as a product of powers of prime factors. Then write it as 2^m x 3^n. (4 marks) 12. Draw a flowchart that shows how to write a large number such as 5,900,000,000 in standard form, step by step. (3 marks) 13. Draw a place value chart for 47561 and below it write each digit multiplied by the correct power of 10. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Simplify and find the value: (i) (2^5 x 7^3) / (8^2 x 7) (ii) (25 x 5^2 x t^8) / (10^3 x t^4) (iii) (3^5 x 10^5 x 25) / (5^7 x 6^5). Show each step. (5 marks) 15. Identify the greater number in each pair and justify: (i) 4^3 or 3^4 (ii) 5^3 or 3^5 (iii) 2^8 or 8^2 (iv) 100^2 or 2^100. What general idea do you notice? (5 marks) 16. The mass of the Earth is 5.97 x 10^24 kg and the mass of the Moon is 7.35 x 10^22 kg. (i) Write both with the same power of 10. (ii) Find their total mass in standard form. (iii) Roughly how many times heavier is the Earth than the Moon? (iv) Why is standard form useful in science? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show which law of exponents you use at each step of simplification.