Mixed practice: Power Play
Power Play24 min readLesson 5 of 7
5.1 Set A: exponent laws
Question 1. Simplify (3^5 × 3^2) ÷ 3^4.
Answer. Group the numerator first: 3^5 × 3^2 = 3^7. Then divide: 3^7 ÷ 3^4 = 3^3 = 27.
Question 2. Find x if 2^x = 128.
Answer. Write 128 as a power of 2: 128 = 2^7, so x = 7. Check: 2^7 = 128.
Question 3. Write 4^3 × 5^3 as a single power and evaluate it.
Answer. Both factors have exponent 3, so 4^3 × 5^3 = (4 × 5)^3 = 20^3 = 8000.
Question 4. Simplify (5^(-2) × 5^5) ÷ 5^2.
Answer. Add the exponents in the numerator: -2 + 5 = 3, giving 5^3. Subtract the denominator exponent: 3 - 2 = 1, so the value is 5.
5.2 Set B: units digits
Question 5. What is the units digit of 6^25?
Answer. Every positive power of 6 ends in 6, so the units digit is 6.
Question 6. What is the units digit of 9^123?
Answer. The units digits of powers of 9 alternate 9, 1, 9, 1, with odd exponents giving 9. Since 123 is odd, the units digit is 9.
Question 7. What is the units digit of 7^2024?
Answer. The units digits of powers of 7 follow the cycle 7, 9, 3, 1, which has period 4. Divide 2024 by 4: 2024 = 4 × 506, so the cycle ends exactly; the units digit is 1.
Question 8. What is the units digit of 13^17?
Answer. Only the units digit of the base matters, so follow powers of 3: the cycle is 3, 9, 7, 1. Since 17 = 4 × 4 + 1, the first member of the cycle applies, so the units digit is 3.
5.3 Set C: scientific notation and scale
Question 9. Write 0.00000407 and 82,600,000 in scientific notation.
Answer. Move the decimal point after the first non-zero digit. For 0.00000407 the point moves six places right, so the form is 4.07 × 10^(-6). For 82,600,000 the point moves seven places left, giving 8.26 × 10^7.
Question 10. A sheet is 0.004 cm thick. Find the thickness after 12 folds.
Answer. Each fold doubles the thickness, so multiply by 2^12 = 4096. The thickness is 0.004 × 4096 = 16.384 cm.
Question 11. Multiply 0.00072 by 3 × 10^4 and write the answer in standard form.
Answer. 0.00072 = 7.2 × 10^(-4). Multiplying the coefficients gives 21.6, and adding the exponents gives 10^0, so the product is 21.6 = 2.16 × 10^1.
Question 12. How many digits does 2^30 have?
Answer. 2^10 = 1024 is a little more than 10^3, so 2^30 = (2^10)^3 is a little more than 10^9. Indeed 2^30 = 1,073,741,824, which has ten digits.
5.4 Set D: growth comparisons
Question 13. Meera's plant grows 3 cm every week. Ravi's plant starts at 2 cm and doubles its height every week. Compare their heights after 6 weeks.
Answer. Meera's plant grows by addition: 3 × 6 = 18 cm. Ravi's plant grows by multiplication: 2 × 2^6 = 2 × 64 = 128 cm. Ravi's plant is far taller because doubling outruns a fixed weekly increase.
Question 14. Which is larger, 2^60 or 3^40?
Answer. Rewrite both with the same exponent: 2^60 = (2^3)^20 = 8^20, and 3^40 = (3^2)^20 = 9^20. Since 9 > 8, 3^40 is larger.
Question 15. A rumour reaches 3 people in the first hour, and each informed person tells 3 new people in the next hour. How many people know after 5 hours if nobody hears it twice?
Answer. The counts are 3, 3^2, 3^3, 3^4, 3^5. After 5 hours, 3^5 = 243 people know. This is exponential growth with base 3.
5.5 Set E: estimates and mixed reasoning
Question 16. Estimate the number of digits in 2^100 without computing it directly.
Answer. Since 2^10 is a little above 10^3, 2^100 = (2^10)^10 is a little above (10^3)^10 = 10^30. A number just above 10^30 has 31 digits. The exact value begins with 1, so the estimate is close.
Question 17. A colony of 500 bacteria doubles every 3 days. How many bacteria are there after 30 days?
Answer. Thirty days contain 10 doubling periods. The count is 500 × 2^10 = 500 × 1024 = 512,000 bacteria.
Question 18. Which is larger, 10^6 or 2^20?
Answer. Write 2^20 = (2^10)^2. Since 2^10 = 1024, 2^20 = 1024^2 = 1,048,576, which is larger than 10^6 = 1,000,000. The power of 2 wins by a small margin.
5.6 Method notes
For exponent questions, write the base and the operation before touching the exponents. Add exponents when multiplying powers of the same base, subtract when dividing, and multiply when one power is raised to another. For units digits, reduce the base to its units digit and find the cycle length before dividing the exponent. For scientific notation, the coefficient stays between 1 and 10, and the exponent counts the decimal places moved. For growth questions, decide whether the change is additive or multiplicative before calculating; the two models look similar in the first step and very different after several steps.