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Figure It Out: Pages 26-27

April 10, 2024
2 min read
solutions scientific-notation simplification

Part 1 (Page 26)

1. Find out the units digit in the value of 2224÷4322^{224} \div 4^{32}.

Solution: First, simplify the expression to a single base. We know 4=224 = 2^2, so 432=(22)32=2644^{32} = (2^2)^{32} = 2^{64}. Expression: 2224÷264=2224−64=21602^{224} \div 2^{64} = 2^{224-64} = 2^{160}.

Now, look at the pattern of units digits for powers of 2:

  • 21=22^1 = 2
  • 22=42^2 = 4
  • 23=82^3 = 8
  • 24=162^4 = 16 (ends in 6)
  • 25=322^5 = 32 (ends in 2 again)

The pattern repeats every 4 powers: 2, 4, 8, 6. Divide exponent 160 by 4: 160÷4=40160 \div 4 = 40 (remainder 0). Remainder 0 corresponds to the 4th position in the cycle, which is 6. Answer: The units digit is 6.

2. There are 5 bottles in a container. Every day, a new container is brought in. How many bottles after 40 days?

Solution: This describes linear growth (adding containers).

  • Day 1: 5 bottles.
  • Day 2: 10 bottles.
  • Day 40: 40×5=20040 \times 5 = 200 bottles.

3. Write the given number as the product of two or more powers in three different ways. (i) 64364^3

  • Way 1: (26)3=218(2^6)^3 = 2^{18}
  • Way 2: (43)3=49(4^3)^3 = 4^9
  • Way 3: (82)3=86(8^2)^3 = 8^6

4. True/False Explanations: (i) Cube numbers are also square numbers. Sometimes True (e.g., 64=43=8264 = 4^3 = 8^2). (ii) Fourth powers are also square numbers. Always True. x4=(x2)2x^4 = (x^2)^2. (v) q46q^{46} is both a 4th power and a 6th power. False. 46 is not divisible by 4 or 6.

5. Simplify: (i) 10−2×10−5=10−710^{-2} \times 10^{-5} = 10^{-7} (ii) 57÷54=535^7 \div 5^4 = 5^3 (iv) (13−2)−3=13(−2×−3)=136(13^{-2})^{-3} = 13^{(-2 \times -3)} = 13^6

6. If 122=14412^2 = 144 what is (1.2)2(1.2)^2? Solution: (1.2)2=(1210)2=144100=1.44(1.2)^2 = (\frac{12}{10})^2 = \frac{144}{100} = 1.44. Similarly, (0.12)2=0.0144(0.12)^2 = 0.0144.

Part 2 (Page 27)

7. Circle numbers that are the same.

  • 24×36=16×7292^4 \times 3^6 = 16 \times 729
  • 64×32=(2×3)4×32=24×34×32=24×366^4 \times 3^2 = (2 \times 3)^4 \times 3^2 = 2^4 \times 3^4 \times 3^2 = 2^4 \times 3^6. (Same as first).
  • 182×62=(18×6)2=108218^2 \times 6^2 = (18 \times 6)^2 = 108^2. Also 182×62=(2×32)2×(2×3)2=22×34×22×32=24×3618^2 \times 6^2 = (2 \times 3^2)^2 \times (2 \times 3)^2 = 2^2 \times 3^4 \times 2^2 \times 3^2 = 2^4 \times 3^6. (Same). Answer: 24×362^4 \times 3^6, 64×326^4 \times 3^2, and 182×6218^2 \times 6^2 are all equal.

9. A dairy plans to produce 8.5 billion packets. How many digits for a unique ID? Solution: 8.5 billion = 8.5×109=8,500,000,0008.5 \times 10^9 = 8,500,000,000. This is a 10-digit number. To cover all these items with a unique ID, we need at least 10 digits (since 1010=10 billion10^{10} = 10 \text{ billion}).

13. Scientific Notation Calculations: (iii) Bacterial cells: 38 trillion = 38×1012=3.8×101338 \times 10^{12} = 3.8 \times 10^{13}. Population of world ≈8 billion=8×109\approx 8 \text{ billion} = 8 \times 10^9. Total bacteria = (3.8×1013)×(8×109)=30.4×1022=3.04×1023(3.8 \times 10^{13}) \times (8 \times 10^9) = 30.4 \times 10^{22} = 3.04 \times 10^{23}.