Question

Find the value of p if (a) 2^5p÷2^p=5(√2^20) (b)( 27)^p =9÷3^p

Answer 1

(a) 2 ^5p ÷ 2 ^p = 5(√2 ^20)

2^5p/2^p = 5(√2 ^20)

we know that

a^m ÷ a^n = a^(m - n)
√2 = 2 ^(1/2)

so
2 ^ (5p - p) = 5 ( 2^ (20×1/2))
2^4p = 5( 2^(10))
p = 5( 2^10 /2^4)
p = 5(2 ^ (10 - 6))
p = 5( 2 ^ 4)
p = 5(16)
p = 80...


(b) 27 ^p = 9 ÷ 3 ^p

27 can be written as 3×3×3...3^3

(3^3)^p = 9 ÷ 3^p

we know that (a ^m)n = a^mn

so 3^3p = 9 / 3^p

9 can be written as 3×3...3^2

3^3p = 3 ^2/3^p

3^3p = 3 ^(2-p)

bases are equal ..so equate the powers

3p = 2 - p
4p = 2
p =2/4 =1/2 is the answer...