Question 7 of 28intermediate🔍 AnalyzeShort Answer2 marks

Circle the numbers that are the same— 24×362^4 \times 3^6 64×326^4 \times 3^2 6106^{10} 182×6218^2 \times 6^2 6246^{24}

Correct Answer

Step 1: Express each in terms of prime factors 2 and 3.

Step 2: 24×362^4 \times 3^6 stays as 24×362^4 \times 3^6.

Step 3: 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

Step 4: 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

Step 5: 610=210×3106^{10} = 2^{10} \times 3^{10} which is different.

Step 6: 6246^{24} is much larger.

Answer: 24×362^4 \times 3^6, 64×326^4 \times 3^2, and 182×6218^2 \times 6^2 are all equal to 64×32=1296×9=116646^4 \times 3^2 = 1296 \times 9 = 11664.

Exercise: Figure it Out (Section 2) | Q: 7 | (Page Page 27)
For More Understanding

Explanation

Convert each expression to prime factors. 24×36=24×362^4 \times 3^6 = 2^4 \times 3^6. 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. 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. 610=210×3106^{10} = 2^{10} \times 3^{10} which is different. 6246^{24} is much larger.

Solution Steps

  1. Express each in terms of prime factors 2 and 3

  2. 24×362^4 \times 3^6 stays as is

  3. 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

  4. 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

  5. 610=210×3106^{10} = 2^{10} \times 3^{10} which is different

  6. 6246^{24} is much larger

  7. The same numbers are: 24×362^4 \times 3^6, 64×326^4 \times 3^2, and 182×6218^2 \times 6^2

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