Question

\( 18^{6}=3^{6}\times 2^{x}\times 6^{y} \)

Ask by Lewis Sanders. in Iran
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\( x = 0 \) and \( y = 6 \).

Solution

We start with the equation \[ 18^6 = 3^6 \times 2^x \times 6^y. \] Notice that \[ 18 = 2 \times 3^2, \] so \[ 18^6 = (2 \times 3^2)^6 = 2^6 \times 3^{12}. \] Also, since \[ 6 = 2 \times 3, \] we have \[ 6^y = (2 \times 3)^y = 2^y \times 3^y. \] Thus, the right-hand side becomes \[ 3^6 \times 2^x \times (2^y \times 3^y) = 2^{x+y} \times 3^{6+y}. \] Now, equate the exponents for the bases 2 and 3: 1. For base 2: \[ 6 = x+y. \] 2. For base 3: \[ 12 = 6+y. \] Solving the equation for \(y\): \[ 12 = 6+y \quad \Rightarrow \quad y=6. \] Now substitute \(y=6\) into \(6=x+y\): \[ 6 = x+6 \quad \Rightarrow \quad x=0. \] Thus, the values are: \[ x=0,\quad y=6. \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Bonus Knowledge

To solve the equation \( 18^{6} = 3^{6} \times 2^{x} \times 6^{y} \), let's start by breaking down \( 18 \): \[ 18 = 2 \times 3^{2} \] Now, we can express \( 18^{6} \): \[ 18^{6} = (2 \times 3^{2})^{6} = 2^{6} \times 3^{12} \] Next, let's rewrite \( 6^{y} \): \[ 6 = 2 \times 3 \] \[ 6^{y} = (2 \times 3)^{y} = 2^{y} \times 3^{y} \] Now, substituting this into our equation gives us: \[ 2^{6} \times 3^{12} = 3^{6} \times 2^{x} \times (2^{y} \times 3^{y}) \] Simplifying the right side: \[ 2^{6} \times 3^{12} = 2^{x+y} \times 3^{6+y} \] Now we can equate the coefficients of \( 2 \) and \( 3 \): For \( 2 \): \[ 6 = x + y \] For \( 3 \): \[ 12 = 6 + y \] From the second equation, we have: \[ y = 6 \] Substituting \( y = 6 \) into the first equation: \[ 6 = x + 6 \] \[ x = 0 \] Thus, the solution is: \[ x = 0, \quad y = 6 \]

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy