Exponents and Powers

Exponents give a short way of writing repeated multiplication, so 2 x 2 x 2 x 2 x 2 is written 2^5, with 2 as the base and 5 as the exponent. In this chapter you will extend exponents to negative integers, learning that x^-m = 1/x^m for a non-zero x, so that 10^-2 = 1/100 and 2^-3 = 1/8. You will see that the laws of exponents, such as x^m x x^n = x^(m+n), x^m / x^n = x^(m-n), (x^m)^n = x^(mn) and x^0 = 1, continue to hold for negative exponents, and you will use them to simplify expressions and solve for unknown exponents. The chapter then uses negative exponents to write very small numbers in standard form, as k x 10^n with k between 1 and 10. The diameter of a red blood cell, about 0.000007 m, becomes 7 x 10^-6 m, while the diameter of the Sun is about 1.4 x 10^9 m. You will convert between standard and usual forms and compare very large and very small quantities.

Detailed Worksheet: Exponents and Powers Section A - Definitions (10 marks) 1. In the expression 7^4, identify the base and the exponent, and write it as a product. (2 marks) 2. What is the meaning of a negative exponent? Write 5^-3 and 10^-4 as fractions. (2 marks) 3. State any four laws of exponents for a non-zero base and integer exponents. (2 marks) 4. What is meant by writing a number in standard form? Write 0.00045 in standard form. (2 marks) 5. Using the law x^m / x^n = x^(m-n), explain why x^0 = 1 for any non-zero x. (2 marks) Section B - Calculations and Applications (15 marks) 6. Evaluate: (i) 3^-2 (ii) (-4)^-2 (iii) (1/2)^-5. (3 marks) 7. Simplify: (i) (-4)^5 / (-4)^8 (ii) (3^-1 + 4^-1 + 5^-1)^0 (iii) (2^-1 x 4^-1) / 2^-2. (3 marks) 8. Find the value of m in each case: (i) 5^m / 5^-3 = 5^5 (ii) (-3)^(m+1) x (-3)^5 = (-3)^7. (3 marks) 9. Express in standard form: (i) 0.0000000000085 (ii) 0.00000000000942 (iii) 6020000000000000. (3 marks) 10. Express in usual form: (i) 3.02 x 10^-6 (ii) 4.5 x 10^4 (iii) 3 x 10^-8. (3 marks) Section C - Diagrams (10 marks) 11. Draw a vertical "powers of ten" scale from 10^-6 m to 10^10 m. Mark on it the approximate position of: the diameter of a red blood cell (7 x 10^-6 m), the thickness of a sheet of paper (1.6 x 10^-5 m), the height of a person (1.6 m), the height of Mount Everest (8.8 x 10^3 m), the diameter of the Earth (1.28 x 10^7 m) and the diameter of the Sun (1.4 x 10^9 m). (4 marks) 12. Draw a pattern chart starting from 2^3 = 8 and dividing by 2 at each step down to 2^-3, showing each step with an arrow labelled "divide by 2". Use the chart to explain the meaning of zero and negative exponents. (3 marks) 13. Draw a place value chart from thousands to thousandths and write the power of 10 for each place above it. Use it to write 1425.36 in expanded form using exponents. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. The diameter of the Sun is 1.4 x 10^9 m and the diameter of the Earth is 1.2756 x 10^7 m. About how many times the Earth's diameter is the Sun's diameter? 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. Find their total mass in standard form. (5 marks) 15. A student wrote: (i) 2^-3 = -8 (ii) 3^2 x 3^-2 = 0 (iii) (2^3)^2 = 2^9 (iv) 5^0 x 5 = 0. Find and correct the mistake in each, naming the law that was misused. (5 marks) 16. In a stack there are 5 books, each 20 mm thick, and 5 paper sheets, each 0.016 mm thick. Find the total thickness of the stack in standard form. The distance from the Earth to the Sun is 1.496 x 10^11 m and from the Earth to the Moon is 3.84 x 10^8 m. During a solar eclipse the Moon is between the Earth and the Sun; find the distance between the Moon and the Sun at that time. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show each law of exponents used, and give answers in standard form wherever asked.