To find the half-life of uranium-238, we start with the given half-life of uranium-235, which is 700 million years. The half-life of uranium-238 is stated to be \( 4.463 \times 10^{11} \) years longer than that of uranium-235.
First, convert 700 million years into the same unit as \( 4.463 \times 10^{11} \) years.
700 million years is the same as: \[ 700 \text{ million} = 700 \times 10^6 \text{ years} = 7.0 \times 10^8 \text{ years} \]
Next, we will add the two half-lives together to determine the half-life of uranium-238: \[ \text{Half-life of uranium-238} = 7.0 \times 10^8 \text{ years} + 4.463 \times 10^{11} \text{ years} \]
To perform the addition, we need both terms in the same magnitude. Let's express \( 7.0 \times 10^8 \) in terms of \( 10^{11} \): \[ 7.0 \times 10^8 = 0.0070 \times 10^{11} \]
Now we can add the two: \[ 0.0070 \times 10^{11} + 4.463 \times 10^{11} = (0.0070 + 4.463) \times 10^{11} = 4.470 \times 10^{11} \text{ years} \]
Thus, the half-life of uranium-238 is approximately: \[ \boxed{4.47 \times 10^{11}} \text{ years} \] When written in decimal form, that is: \[ \text{Half-life of uranium-238} \approx 447,000,000,000 \text{ years} \]
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