LCM of 81a^2b^2 and 27a^3b^3
// FACTOR BREAKDOWN
81 = 3 x 3 x 3 x 3
27 = 3 x 3 x 3
a^3 = a x a x a
a^2 = a x a
b^3 = b x b x b
b^2 = b x b
/* For LCM, we want to consider the most number of times any digit or variable appears in the factorization. Comparing 81 and 27, we see that the number three shows up four times in the factorization, so the LCM of these two numbers is 81. Alternatively, you can see that 27 can go into 81 three times.
For variables, it's the same concept. We have three a's appear so that is our most number of times we see a.
Same logic for the variable b. */
//FINAL ANSWER
81a^3b^3
Ok thank you so much for your time! I got a bit confused on the multiplying of LCM but now I understand.
To find the least common multiple (LCM) of the given expressions, we need to first factorize the expressions completely.
Let's start with the first expression: 81a^2b^2.
Factorizing 81, we get: 81 = 3^4.
Next, let's factorize a^2: a^2 = a * a.
And finally, let's factorize b^2: b^2 = b * b.
Putting it all together, we have: 81a^2b^2 = (3^4)(a * a)(b * b).
Now, let's move on to the second expression: 27a^3b^3.
Factorizing 27, we get: 27 = 3^3.
Next, let's factorize a^3: a^3 = a * a * a.
And finally, let's factorize b^3: b^3 = b * b * b.
Putting it all together, we have: 27a^3b^3 = (3^3)(a * a * a)(b * b * b).
Now, to find the LCM, we need to take the highest power of each factor that appears in both expressions.
In this case, we have the factor 3 appearing with a power of 4 in the first expression and a power of 3 in the second expression. So, we take the highest power, which is 4.
The factor a appears with a power of 2 in the first expression and a power of 3 in the second expression. So, we take the highest power, which is 3.
The factor b appears with a power of 2 in the first expression and a power of 3 in the second expression. So, we take the highest power, which is 3.
Putting it all together, the LCM of 81a^2b^2 and 27a^3b^3 is: (3^4)(a^3)(b^3) = 81a^3b^3.