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What is the pattern in powers of 2 and how do you calculate 26 and 23 x 24?

Mathematics

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Step-by-step solution

10 steps

1.1.1

  1. Observe the values in the table for 2n2^n

    2, 4, 8, 16, 32.

  2. Notice that each value is obtained by multiplying the previous value by 2.

    The pattern is that each successive value of 2n2^n is double the previous value.

    1.1.2

  3. The pattern observed is that each term is multiplied by 2 to get the next term.

  4. The value for 252^5 is 32.

    To predict 262^6, multiply 252^5 by 2.

    26=32×2=642^6 = 32 \times 2 = \boxed{\text{64}}

    My reasoning is that the pattern shows each power of 2 is twice the previous power of 2.

    1.1.2 (Verification)

  5. Calculate 262^6 by multiplying 2 by itself 6 times.

    26=2×2×2×2×2×22^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2

  6. Perform the multiplication.

    26=4×2×2×2×2=8×2×2×2=16×2×2=32×2=642^6 = 4 \times 2 \times 2 \times 2 \times 2 = 8 \times 2 \times 2 \times 2 = 16 \times 2 \times 2 = 32 \times 2 = \boxed{\text{64}}

    1.2.1

  7. Write each power as repeated multiplication.

    23=2×2×22^3 = 2 \times 2 \times 2 24=2×2×2×22^4 = 2 \times 2 \times 2 \times 2

  8. Combine the repeated multiplications and calculate the product.

    23×24=(2×2×2)×(2×2×2×2)2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) 23×24=8×16=1282^3 \times 2^4 = 8 \times 16 = \boxed{\text{128}}

    1.2.2.1

  9. Sipho's answer is 272^7.

    Calculate 272^7.

    27=2×2×2×2×2×2×2=1282^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128

  10. Compare Sipho's result with the calculated answer from 1.2.1.

    Sipho's answer (27=1282^7 = 128) matches the calculated answer (128).
    Sipho is Correct\boxed{\text{Correct}}.

    1.2.2.2
    Sipho arrived at 7 by adding the exponents of the original terms, 3 and 4, since 3+4=73+4=7.

    1.2.3
    The general law for the given calculations is:

    am×an=am+na^m \times a^n = \boxed{a^{m+n}}
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