Chapter Overview
\(10^{24}\)
Mass of Earth Example
\(a^{-m}\)
Negative Exponent
5
Laws of Exponents
\(10^{-6}\)
Standard Form Small Numbers
What You'll Learn
Introduction to Exponents
Understanding how to express large numbers using exponents like \(5.97 \times 10^{24}\).
Negative Exponents
Learning powers with negative exponents and their relation to reciprocals.
Laws of Exponents
Exploring laws for multiplication, division, and powers with integers.
Standard Form
Expressing small and large numbers in scientific notation and comparisons.
Key Concepts Summary
This chapter introduces exponents for large numbers, such as the mass of Earth as \(5.97 \times 10^{24}\) kg. It covers negative exponents, where \(10^{-1} = \frac{1}{10}\), and extends laws of exponents to negative cases. Standard form is used for small numbers like 0.000007 m as \(7 \times 10^{-6}\) m, with comparisons between large and small values.
Key Highlights
Exponents simplify large and small numbers. Laws include \(a^m \times a^n = a^{m+n}\) and \(a^{-m} = \frac{1}{a^m}\). Applications include scientific notation for distances, sizes, and masses, enabling easy comparisons like Sun's diameter being about 100 times Earth's.
Comprehensive Chapter Summary
1. Introduction to Exponents
The chapter begins with expressing large numbers using exponents, like the mass of Earth as \(5.97 \times 10^{24}\) kg. It explains \(10^{24}\) as 10 raised to the power 24, and reviews positive exponents like \(2^5 = 2 \times 2 \times 2 \times 2 \times 2\).
2. Powers with Negative Exponents
Definition and Patterns
Negative exponents are introduced with patterns: \(10^{-1} = \frac{1}{10}\), \(10^{-2} = \frac{1}{100}\). Generally, \(a^{-m} = \frac{1}{a^m}\) for non-zero integer a and positive m.
Multiplicative Inverse
\(a^{-m}\) is the multiplicative inverse of \(a^m\). Examples include finding inverses like \(2^{-4} = \frac{1}{16}\).
Expanded Form with Decimals
Numbers like 1425.36 are expanded as \(1 \times 10^3 + 4 \times 10^2 + 2 \times 10^1 + 5 \times 10^0 + 3 \times 10^{-1} + 6 \times 10^{-2}\).
3. Laws of Exponents
Extension to Negative Exponents
Laws hold for integers: \(a^m \times a^n = a^{m+n}\), verified with examples like \(2^{-3} \times 2^{-2} = 2^{-5}\).
Key Laws
(i) \(\frac{a^m}{a^n} = a^{m-n}\), (ii) \((a^m)^n = a^{mn}\), (iii) \(a^m \times b^m = (ab)^m\), (iv) \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\), (v) \(a^0 = 1\).
Examples and Simplifications
Examples include simplifying \((-4)^5 \times (-4)^{-10} = (-4)^{-5}\), and finding m in equations like \((-3)^{m+1} \times (-3)^5 = (-3)^7\).
4. Standard Form for Small Numbers
Expression and Facts
Small numbers like 0.000007 = \(7 \times 10^{-6}\). Facts include sizes of cells, thicknesses, and distances in standard form.
5. Comparing Large and Small Numbers
Comparisons
Sun's diameter is about 100 times Earth's: \(\frac{1.4 \times 10^9}{1.2756 \times 10^7} \approx 100\). Red blood cell is half plant cell size.
Addition and Subtraction
Total mass of Earth and Moon: \(5.97 \times 10^{24} + 7.35 \times 10^{22} = 6.0435 \times 10^{24}\). Sun-Moon distance during eclipse.
Additional Formulas and Content
More formulas: For any a, \(a^m \div a^n = a^{m-n}\), extended to negatives. Example: \(10^{-3} = \frac{1}{1000}\). Comparisons like speed of light \(3 \times 10^8\) m/s vs. cell size \(10^{-5}\) m.
Questions and Answers from Chapter
Short Questions
Q1. Evaluate \(3^{-2}\).
Answer: \(\frac{1}{9}\)
Q2. Evaluate \((-4)^{-2}\).
Answer: \(\frac{1}{16}\)
Q3. Evaluate \(\left(\frac{1}{2}\right)^{-5}\).
Answer: \(32\)
Q4. Simplify \((-4)^5 \div (-4)^8\).
Answer: \((-4)^{-3}\)
Q5. Simplify \(\left(2^3\right)^2\).
Answer: \(2^6\)
Q6. Find multiplicative inverse of \(2^{-4}\).
Answer: \(16\)
Q7. Find multiplicative inverse of \(10^{-5}\).
Answer: \(100000\)
Q8. Evaluate \( (3^0 + 4^{-1}) \times 2^2 \).
Answer: \(5\)
Q9. Evaluate \( (2^{-1} \times 4^{-1}) \div 2^{-2} \).
Answer: \(\frac{1}{2}\)
Q10. Evaluate \(\left(\frac{1}{2}\right)^2 + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{4}\right)^2\).
Answer: \(\frac{49}{144}\)
Q11. Evaluate \((3^{-1} + 4^{-1} + 5^{-1})^0\).
Answer: \(1\)
Q12. Evaluate \(\frac{8^{-1} \times 5^3}{2^{-4}}\).
Answer: \(250\)
Q13. Evaluate \((5^{-1} \times 2^{-1}) \times 6^{-1}\).
Answer: \(\frac{1}{60}\)
Q14. Find m for \(5^m \div 5^{-3} = 5^5\).
Answer: \(8\)
Q15. Simplify \(\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}}\).
Answer: \(5 t^4\)
Medium Questions
Q1. Simplify and express in power notation with positive exponent: \(\left( \frac{1}{2^3} \right)^2\).
Answer: \(\frac{1}{64} = 2^{-6}\) (3 marks)
Q2. Simplify and express in power notation with positive exponent: \((-4)^5 \div (-4)^8\).
Answer: \((-4)^{-3} = \frac{1}{(-4)^3}\) (3 marks)
Q3. Simplify and express in power notation with positive exponent: \((3^{-7} \div 3^{-10}) \times 3^{-5}\).
Answer: \(3^{-2} = \frac{1}{9}\) (3 marks)
Q4. Simplify and express in power notation with positive exponent: \(2^{-3} \times (-7)^{-3}\).
Answer: \((-14)^{-3} = \frac{1}{(-14)^3}\) (3 marks)
Q5. Evaluate \(\left\{ \left( \frac{1}{3} \right)^{-1} - \left( \frac{1}{4} \right)^{-1} \right\}^{-1}\).
Answer: \(-12\) (3 marks)
Q6. Evaluate \(\frac{5^7}{8} \times \left( \frac{8}{5} \right)^{-7}\).
Answer: \(5^{14}\) (3 marks)
Q7. Simplify \(\frac{125 \times 5^{-5} \times 6^{-3}}{5^{-7} \times 6^{-5}}\).
Answer: \(6^2 \times 5^7\) (3 marks)
Q8. Express 0.0000000000085 in standard form.
Answer: \(8.5 \times 10^{-12}\) (3 marks)
Q9. Express 6020000000000000 in standard form.
Answer: \(6.02 \times 10^{15}\) (3 marks)
Q10. Express 3.02 × 10^{-6} in usual form.
Answer: 0.00000302 (3 marks)
Q11. Express 1 micron (\(\frac{1}{1000000}\) m) in standard form.
Answer: \(10^{-6}\) m (3 marks)
Q12. Find total thickness of 5 books (20mm each) and 5 sheets (0.016 mm each).
Answer: 100.08 mm (3 marks)
Q13. Expand 1025.63 using exponents.
Answer: \(1 \times 10^3 + 0 \times 10^2 + 2 \times 10^1 + 5 \times 10^0 + 6 \times 10^{-1} + 3 \times 10^{-2}\) (3 marks)
Q14. Simplify \((-2)^{-3} \times (-2)^{-4}\).
Answer: \((-2)^{-7}\) (3 marks)
Q15. Simplify \(p^3 \times p^{-10}\).
Answer: \(p^{-7}\) (3 marks)
Long Questions
Q1. Simplify and express the result in power notation with positive exponent: \(( -4 )^5 \div ( -4 )^8\).
Answer: Using the law \(a^m \div a^n = a^{m-n}\), \((-4)^5 \div (-4)^8 = (-4)^{5-8} = (-4)^{-3}\). To express with positive exponent, \((-4)^{-3} = \frac{1}{(-4)^3} = -\frac{1}{64}\).
Q2. Simplify and express the result in power notation with positive exponent: \((3^{-7} \div 3^{-10}) \times 3^{-5}\).
Answer: First, \(3^{-7} \div 3^{-10} = 3^{-7 - (-10)} = 3^3 = 27\). Then, \(27 \times 3^{-5} = 3^3 \times 3^{-5} = 3^{-2} = \frac{1}{9}\).
Q3. Find the value of: \((3^0 + 4^{-1}) \times 2^2\).
Answer: \(3^0 = 1\), \(4^{-1} = \frac{1}{4}\), so \(1 + \frac{1}{4} = \frac{5}{4}\). Then, \(\frac{5}{4} \times 4 = 5\).
Q4. Find the value of: \((2^{-1} \times 4^{-1}) \div 2^{-2}\).
Answer: \(2^{-1} = \frac{1}{2}\), \(4^{-1} = \frac{1}{4}\), so \(\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\). Then, \(\frac{1}{8} \div \frac{1}{4} = \frac{1}{8} \times 4 = \frac{1}{2}\).
Q5. Find the value of: \((3^{-1} + 4^{-1} + 5^{-1})^0\).
Answer: Any non-zero number to the power 0 is 1. Here, \(3^{-1} + 4^{-1} + 5^{-1} = \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = \frac{20 + 15 + 12}{60} = \frac{47}{60} \neq 0\), so \( \left( \frac{47}{60} \right)^0 = 1 \).
Q6. Evaluate: \(\frac{8^{-1} \times 5^3}{2^{-4}}\).
Answer: \(8^{-1} = \frac{1}{8}\), \(5^3 = 125\), \(2^{-4} = \frac{1}{16}\). So, \(\frac{\frac{1}{8} \times 125}{\frac{1}{16}} = \frac{125}{8} \times 16 = \frac{125 \times 16}{8} = 125 \times 2 = 250\).
Q7. Evaluate: \((5^{-1} \times 2^{-1}) \times 6^{-1}\).
Answer: \(5^{-1} = \frac{1}{5}\), \(2^{-1} = \frac{1}{2}\), \(6^{-1} = \frac{1}{6}\). So, \(\frac{1}{5} \times \frac{1}{2} \times \frac{1}{6} = \frac{1}{60}\).
Q8. Find m for which \(5^m \div 5^{-3} = 5^5\).
Answer: \(5^m \div 5^{-3} = 5^{m - (-3)} = 5^{m+3} = 5^5\). So, m + 3 = 5, m = 2.
Q9. Evaluate \(\left\{ \left( \frac{1}{3} \right)^{-1} - \left( \frac{1}{4} \right)^{-1} \right\}^{-1}\).
Answer: \(\left( \frac{1}{3} \right)^{-1} = 3\), \(\left( \frac{1}{4} \right)^{-1} = 4\), so 3 - 4 = -1. Then, \((-1)^{-1} = -1\).
Q10. Simplify \(\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}}\) (t ≠ 0).
Answer: \(25 = 5^2\), 10 = 2 \times 5, so \(\frac{5^2 \times t^{-4}}{5^{-3} \times 2 \times 5 \times t^{-8}} = 5^{2 - (-3) - 1} \times \frac{1}{2} \times t^{-4 - (-8)} = \frac{5^4}{2} \times t^4 = \frac{625 t^4}{2}\).
Q11. Simplify \(\frac{125 \times 5^{-5} \times 6^{-3}}{5^{-7} \times 6^{-5}}\).
Answer: 125 = 5^3, so \(\frac{5^3 \times 5^{-5} \times 6^{-3}}{5^{-7} \times 6^{-5}} = 5^{3 -5 - (-7)} \times 6^{-3 - (-5)} = 5^{5} \times 6^{2} = 3125 \times 36 = 112500\).
Q12. Express the following numbers in standard form: 0.0000000000085.
Answer: Move decimal 12 places right: 8.5 × 10^{-12}.
Q13. Express the following numbers in usual form: 3.02 × 10^{-6}.
Answer: Move decimal 6 places left: 0.00000302.
Q14. In a stack there are 5 books each of thickness 20mm and 5 paper sheets each of thickness 0.016 mm. What is the total thickness of the stack.
Answer: Books: 5 × 20 = 100 mm. Sheets: 5 × 0.016 = 0.08 mm. Total: 100.08 mm.
Q15. Express the number appearing in the following statements in standard form: Charge of an electron is 0.000,000,000,000,000,000,16 coulomb.
Answer: 1.6 × 10^{-19} coulomb.
