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If 3(K−2)×3K=729, then the value of K is:
2
4
8
6
4
Express 729 as a power of 3 (3⁶) and equate the powers: K - 2 + K = 6, which leads to 2K = 8.
The equation is 3^(K - 2) × 3ᴷ = 729.
am×an=am+n
Express 729 as a power of 3 (3⁶) and equate the powers: K - 2 + K = 6, which leads to 2K = 8.
A common mistake is incorrectly adding or subtracting the exponents, specifically forgetting to combine the K terms (K+K=2K) before solving for K.
Apply Product Rule
Use the exponent property am×an=am+n to combine the terms on the left side.
3(K−2)+K=32K−2
Express as base 3
Write 729 as a power of 3. Since 31=3,32=9,33=27,34=81,35=243,36=729.
729=36
Equate the Exponents
Since the bases are equal, the exponents must be equal: 2K−2=6.
2K−2=6
Solve for K
Add 2 to both sides to get 2K=8, then divide by 2.
K=4
B is correct because solving the equation 32K−2=36 leads to 2K−2=6, so 2K=8, which means K=4.
This concept of equating bases is foundational for solving exponential and logarithmic equations in higher algebra.
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