IF the LCM of two numbers a and b is 40, what is the LCM of 7a^2 and 7b^2?

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To find the LCM of 7a^2 and 7b^2, we need to determine the LCM of the two base numbers - 7 and 7. Since both numbers are the same, the LCM will be the same.

The LCM of two equal numbers is the number itself. Therefore, the LCM of 7 and 7 is 7.

Now, let's consider the exponents a^2 and b^2. Since the exponents are both squared, they are the same, so we can ignore them for finding the LCM.

Therefore, the LCM of 7a^2 and 7b^2 is 7.

To find the LCM of 7a^2 and 7b^2, we need to determine the highest power of each prime factor present in both terms.

First, let's factorize 7a^2 and 7b^2:
7a^2 = 7 * a * a
7b^2 = 7 * b * b

From these factorizations, we can see that both 7a^2 and 7b^2 have the same prime factors, namely 7, a, a, b, and b. We need to determine the highest power of each prime factor in order to find the LCM.

For the prime factor 7, it appears only once in both terms. So, the highest power of 7 we can take is 7.

For the prime factor a, it appears twice in both terms (a * a). Hence, the highest power of a we can take is a^2.

For the prime factor b, it also appears twice in both terms (b * b). Therefore, the highest power of b we can take is b^2.

Now, we can form the LCM of 7a^2 and 7b^2:
LCM(7a^2, 7b^2) = 7 * a^2 * b^2

Since we know that the LCM of a and b is 40 (given in the question), we can substitute this information:
LCM(7a^2, 7b^2) = 7 * a^2 * b^2 = 7 * 40 = 280

Therefore, the LCM of 7a^2 and 7b^2 is 280.