The prime factorization of 243 is 3*3*3*3*3. Write 243 as the product of two powers of 3 in as many ways as possible.
Thank you
my buttcheeks burn from math so this really helps
3^0*3^5 (1*243)
3^1*3^4 (3*81)
3^2*3^3 (9*27)
and in reverse order
You're welcome! Here are the different ways to write 243 as the product of two powers of 3:
1. 243 = 3^5
2. 243 = 3^4 * 3^1
3. 243 = 3^3 * 3^2
To write 243 as the product of two powers of 3, we need to find two exponents that, when multiplied together, equal 243.
Let's start with the prime factorization of 243: 3 * 3 * 3 * 3 * 3.
To express 243 as the product of two powers of 3, we can group the prime factors in different combinations:
1) 3^5 * 3^0 = 3^5
Here, we group all the factors (3 * 3 * 3 * 3 * 3) and combine them. The exponent is 5 since we have 5 factors of 3.
2) 3^4 * 3^1 = 3^5
In this case, we group four factors of 3 together and keep one factor separate. So, the exponent of 3 is 4 + 1 = 5.
3) 3^3 * 3^2 = 3^5
Here, we group three factors of 3 together and two factors separately. The exponent is 3 + 2 = 5.
4) 3^2 * 3^3 = 3^5
This time, we group two factors of 3 together and three factors separately. The exponent is 2 + 3 = 5.
Therefore, 243 can be written as the product of two powers of 3 in four different ways: 3^5, 3^4 * 3^1, 3^3 * 3^2, and 3^2 * 3^3.