Question
upstudy study bank question image url

Given: \( \frac{4^{x-1}+4^{x+1}}{17.12^{x}} \) \( 2.2 .1 \quad \) Simplify the expression fully. \( 2.2 .2 \quad \) If \( 3^{-x}=4 t \), express \( \frac{4^{x-1}+4^{x+1}}{17.12^{x}} \) in terms of \( t \).

Ask by Norris Hampton. in South Africa
Mar 08,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The fully simplified expression is \( \frac{1}{4} \cdot 3^{-x} \). In terms of \( t \), it is \( t \).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To simplify the expression \( \frac{4^{x-1}+4^{x+1}}{17.12^{x}} \), we start by rewriting \( 4^{x-1} \) and \( 4^{x+1} \). We have: \[ 4^{x-1} = \frac{4^x}{4} \quad \text{and} \quad 4^{x+1} = 4 \cdot 4^x. \] Thus, \[ 4^{x-1} + 4^{x+1} = \frac{4^x}{4} + 4 \cdot 4^x = \frac{1}{4}4^x + 4 \cdot 4^x = \frac{1 + 16}{4}4^x = \frac{17}{4}4^x. \] Now substituting this back into the original expression gives: \[ \frac{\frac{17}{4}4^x}{17.12^x} = \frac{17}{4 \cdot 17 \cdot 12^x}4^x = \frac{4^x}{4 \cdot 12^x} = \frac{4^x}{48^x} = \left(\frac{4}{48}\right)^{x} = \left(\frac{1}{12}\right)^{x}. \] Thus, the simplified expression is \( \left(\frac{1}{12}\right)^x \). Now, for the second part, we have \( 3^{-x}=4 t \), which can be rearranged to express \( 3^{x} = \frac{1}{4t} \). Using this, we need to relate it to our simplified form \( \left(\frac{1}{12}\right)^x \). Notice that: \[ 12 = 3 \times 4, \] So, \[ \left(\frac{1}{12}\right)^x = \left(\frac{1}{3 \cdot 4}\right)^x = (3^{-1} \cdot 4^{-1})^x = 3^{-x} \cdot 4^{-x}. \] From \( 3^{-x} = 4t \), we substitute to get: \[ \left(\frac{1}{12}\right)^x = (4t) \cdot 4^{-x} = 4t \cdot \left(\frac{1}{4}\right)^x = 4t \cdot \frac{1}{4^x}. \] Thus, if \( 3^{-x} = 4t \), we express \( 4^{-x} \) as \( (4^{-1})^x \). Since \( 4^{-x} = \left(\frac{1}{4}\right)^x\), we illustrate that: \[ \frac{4^{x-1}+4^{x+1}}{17.12^{x}} = \frac{4t}{4^x}. \] Therefore, the expression in terms of \( t \) is \( 4t \cdot \left(\frac{1}{4}\right)^{x} \).

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