1. If $a \star b = 2a + 3b $ for all a and b, then what is the value of $4 \star 3$?
2. Define $\#N$ by the formula $\#N = .5(N)+1$. Calculate $\#(\#(\#50))$.
3. If $a+b=8$, $b+c=-3$, and $a+c=-5$, what is the value of the product $abc$?
4. Davey Q. decides to factor the number 756 into primes. After doing so, he makes a list from this factorization, including each prime in the list as many times as it appears in the factorization. What is the mode of his list?
P.S: Plz label questions 1-4 when giving answers~
1. a \star b= 2a+3b
We plug in 4 and 3 for a and b. We have:
4 \star 3=2(4)+3(3)
2. We plug in 50 for N in the given equation. Our answer is 8.
3. We can add all equations together to get 2a+2b+2c=0. Then a+b+c=0. c must be -8, a must be 3, and b is 5. The product of abc is -8(3)(5)=-8*15=-120.
4. We prime factorize 756 to get 2^2 • 3^3 • 7. Mode is the number that appears most in the set. Since 3 appears 3 times in the prime factorization, which is the most, 3 is the mode of his list.