Find the least common multiple of the monomials. 5a squared, and 16a cubed. Also 17b squared and 3b cubed. Could you show me how to do these.
Thank you,
Sandy
5a^2 = 5*a^2
16a^2 = 2^4 * a^2
take the highest power of each prime (assuming a is not divisible by 2 or 5)
LCM = 2^4*5*a^2 = 80a^2
17b^2 = 17*b^2
3b^3 = 3*b^3
LCM = 3*17*b^3 = 51b^3
The LCM is the smallest number that can be divided by each of the two given numbers.
another example: LCM(20,30,50,80):
20 = 2^2 * 5
30 = 2 * 3 * 5
50 = 2 * 5^2
80 = 2^4 * 5
LCM = 2^4 * 3 * 5^2 = 1200
To find the least common multiple (LCM) of monomials, you need to find the highest power of each variable that appears in any of the monomials.
For 5a^2 and 16a^3:
- For the variable 'a', the highest power is a^3.
- For the variable '5', there is no power, so it can be considered as a^0.
So the LCM of 5a^2 and 16a^3 is 5a^3.
For 17b^2 and 3b^3:
- For the variable 'b', the highest power is b^3.
- For the constant '17', it can be considered as b^0.
- For the constant '3', it can also be considered as b^0.
So the LCM of 17b^2 and 3b^3 is 3b^3.
Therefore, the LCM of 5a^2 and 16a^3 is 5a^3, and the LCM of 17b^2 and 3b^3 is 3b^3.
Certainly, Sandy! To find the least common multiple (LCM) of monomials, we need to find the highest power of each variable that appears in either monomial and multiply them together.
Let's start with the first set of monomials: 5a^2 and 16a^3.
Step 1: Write down the prime factorization of both exponents:
The prime factorization of 2 (the exponent of 'a^2') is 2^2.
The prime factorization of 3 (the exponent of 'a^3') is 3^1.
Step 2: Identify the highest power of each prime factor:
In this case, the highest power of 2 is 2^2, and the highest power of 3 is 3^1.
Step 3: Multiply the highest powers of each prime factor:
2^2 * 3^1 = 4 * 3 = 12 (this is the LCM of the exponents)
Step 4: Combine the variable with the LCM of the exponents:
The LCM of the monomials 5a^2 and 16a^3 is 12a^3.
Now let's move on to the second set of monomials: 17b^2 and 3b^3.
Step 1: Write down the prime factorization of both exponents:
The prime factorization of 2 (the exponent of 'b^2') is 2^1.
The prime factorization of 3 (the exponent of 'b^3') is 3^1.
Step 2: Identify the highest power of each prime factor:
In this case, the highest power of 2 is 2^1, and the highest power of 3 is 3^1.
Step 3: Multiply the highest powers of each prime factor:
2^1 * 3^1 = 2 * 3 = 6 (this is the LCM of the exponents)
Step 4: Combine the variable with the LCM of the exponents:
The LCM of the monomials 17b^2 and 3b^3 is 6b^3.
Therefore, the LCM of 5a^2 and 16a^3 is 12a^3, and the LCM of 17b^2 and 3b^3 is 6b^3.
Remember, to find the LCM of monomials, you need to find the highest power of each prime factor and multiply them together.