Solve the equation. Round to 4 decimal places when necessary.

Question

(1/9)^k ⋅ 27^k+4 = (1/81)^3-k

Answer 1

(1/9)^k(27)^k+4 = (1/81)^3-k

(3^-2)^k(3³)^k+4 = (3^-4)^3-k

3^-2k3^3k+12 = 3^-12+4k

3^k+12 = 3^-12+4k

k+12 = 4k - 12

-3k = -24

k = 8

Answer 2

I'm going to assume you accidently left out parenthesis for the exponent expressions and this is the equation:

(1/9)^k ⋅ 27^k+4 = (1/81)^3-k

1/9 = 1/3² = 3^-2 and 1/81 = 1/3⁴ = 3^-4 and 27 = 3³ so the equation can be written as:

(3^-2)^k • (3³)^k+4 = (3^-4)^3-k

3^-2k • 3^3k+12 = 3^-12+4k

3^{-2k + 3k+12} = 3^-12+4k

3^k+12 = 3^-12+4k

Therefore k + 12 = -12 + 4k

3k = 24

k = 8

Answer 3

This is a very common exponential equation type: the bases are not = , but all of the terms in the equation can be given a common base. This will allow us to solve most, or all, of the equation by hand (analytically):

Look at the 3 bases given: 1/9, 27, 1/81. They are all powers of 3. Thus, we will write them all as powers of three, apply exponent rules, and then we will be able to ignore the base and set the exponents =.

(3^-2)^k·(3³)^k+4 = (3^-4)^3-k Now use the exponent rule that raising a power to a power means multiply exponents.

3^-2k·3^3k+12 = 3^4k-12 2nd exponent rule: when multiplying expressions with the same base, we add exponents.

3^k+12 = 3^4k-12 Now since the bases are the same on both sides, we ignore them and equate the exponents:

k + 12 = 4k - 12

k = 8. This is a very easy answer to check, so we ought to do that to verify that k = 8 solves the equation ...

Answer 4

To solve this you need to re write all of these expressions as powers of the same base which in this case is 3

(3)^-2k (3^3)^k+4 = (3^-4)^3-k

(3)^-2k (3)^3k+12= (3)^-12+4k

(3)^k+12= (3)^-12+4k

k+ 12= -12 +4k

3k =24

k=8

Answer 5

(1/9)^k27^k+4 = (1/81)^3-k

9, 27 and 81 are all multiples of 3

(1/3²)^k(3³)^k+4 = (1/3⁴)^3-k

3^-2k3^3k+12 = 3^4k-12

^{We can add exponents when multiplying}

3^k+12 = 3^4k-12

Since everything is to the same base of 3, we can equate exponents:

k+12 = 4k-12

3k= 24

k = 24/3 = 8

Answer 6

(1/9)^k • 27^k+4 = (1/81)^3-k

(3^-2)^k • (3³)^k+4 = (3^-4)^3-k

(3)^-2k • (3)^3k+12 = (3)^-12+4k

(3)^-2k+3k+12 = (3)^-12+4k

(3)^k+12 = (3)^-12+4k

k + 12 = -12 + 4k

3k = 24

k = 8