Question
upstudy study bank question image url

4) \( \left(\frac{3 a^{6} b^{5}}{6 a^{7} b^{4}}\right)^{4} \)

Ask by Chadwick Little. in Colombia
Mar 11,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\[ \frac{b^4}{16a^4} \]

Solution

1. Simplificamos la fracción dentro del paréntesis: \[ \frac{3 a^{6} b^{5}}{6 a^{7} b^{4}} = \frac{3}{6}\cdot \frac{a^{6}}{a^{7}}\cdot \frac{b^{5}}{b^{4}} \] 2. Simplificamos cada término por separado: - Coeficientes: \(\frac{3}{6} = \frac{1}{2}\). - Potencias de \(a\): \(\frac{a^{6}}{a^{7}} = a^{6-7} = a^{-1}\). - Potencias de \(b\): \(\frac{b^{5}}{b^{4}} = b^{5-4} = b^1 = b\). 3. Combinamos los resultados obtenidos: \[ \frac{1}{2} \cdot a^{-1} \cdot b = \frac{b}{2a} \] 4. Elevamos el resultado a la cuarta potencia: \[ \left( \frac{b}{2a} \right)^{4} = \frac{b^4}{(2a)^4} \] 5. Simplificamos la potencia del denominador: \[ (2a)^4 = 2^4 \cdot a^4 = 16a^4 \] 6. La expresión final es: \[ \frac{b^4}{16a^4} \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

The Deep Dive

To simplify the expression \( \left(\frac{3 a^{6} b^{5}}{6 a^{7} b^{4}}\right)^{4} \), we start with the fraction inside the parentheses: 1. Simplify the constants: \( \frac{3}{6} = \frac{1}{2} \). 2. For the variables, we apply the laws of exponents: - For \( a \): \( \frac{a^{6}}{a^{7}} = a^{6-7} = a^{-1} \). - For \( b \): \( \frac{b^{5}}{b^{4}} = b^{5-4} = b^{1} = b \). Now we can rewrite the fraction: \[ \frac{3 a^{6} b^{5}}{6 a^{7} b^{4}} = \frac{1}{2} a^{-1} b \] Next, we raise this to the power of 4: \[ \left(\frac{1}{2} a^{-1} b\right)^{4} = \left(\frac{1}{2}\right)^{4} (a^{-1})^{4} b^{4} \] Calculating each part: - \( \left(\frac{1}{2}\right)^{4} = \frac{1}{16} \) - \( (a^{-1})^{4} = a^{-4} \) - \( b^{4} = b^{4} \) Now we combine these results: \[ \frac{1}{16} a^{-4} b^{4} = \frac{b^{4}}{16 a^{4}} \] So the final simplified expression is: \[ \frac{b^{4}}{16 a^{4}} \]

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