Math

Question Compute the product of 3×1053 \times 10^{5} and 7×1027 \times 10^{2}.

Studdy Solution

STEP 1

Assumptions
1. We are performing a multiplication of two numbers in scientific notation.
2. The base for the scientific notation is 10.

STEP 2

Recall the properties of exponents, specifically that when multiplying powers with the same base, you add the exponents.
am×an=am+na^m \times a^n = a^{m+n}

STEP 3

Apply the properties of exponents to the given problem by separating the coefficients (3 and 7) from the powers of 10.
(3×105)(7×102)=3×7×105×102\left(3 \times 10^{5}\right)\left(7 \times 10^{2}\right) = 3 \times 7 \times 10^{5} \times 10^{2}

STEP 4

First, multiply the coefficients (3 and 7).
3×7=213 \times 7 = 21

STEP 5

Next, add the exponents of the powers of 10, since the base is the same (10).
105×102=105+2=10710^{5} \times 10^{2} = 10^{5+2} = 10^{7}

STEP 6

Combine the results from the coefficient multiplication and the exponent addition to get the final answer.
21×10721 \times 10^{7}

STEP 7

Write the final answer in scientific notation.
(3×105)(7×102)=21×107\left(3 \times 10^{5}\right)\left(7 \times 10^{2}\right) = 21 \times 10^{7}
The result of the computation is 21×10721 \times 10^{7}.

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