Understanding the Expression
To simplify the expression √(a^3/4 × b^2/3 × c^4 ÷ 3√(a^6b^-3c^6)), we first need to rewrite it in a more manageable form.
Step 1: Rewrite the Expression
- The square root can be expressed as an exponent:
- √(x) = x^(1/2)
- The cube root can also be expressed similarly:
- 3√(x) = x^(1/3)
This leads to:
√(a^(3/4) × b^(2/3) × c^4) ÷ (a^(6/3) × b^(-3/3) × c^(6/3))^(1/3)
Step 2: Combine the Terms
- The expression can be rewritten as:
(a^(3/4) × b^(2/3) × c^4) / (a^(2) × b^(-1) × c^(2))
Step 3: Apply Exponent Rules
- When dividing like bases, subtract the exponents:
- a^(3/4 - 2) = a^(-5/4)
- b^(2/3 - (-1)) = b^(2/3 + 3/3) = b^(5/3)
- c^(4 - 2) = c^(2)
This results in:
a^(-5/4) × b^(5/3) × c^(2)
Step 4: Final Result
- The final simplified expression is:
b^(5/3) × c^(2) / a^(5/4)
This is your simplified expression in a clear format.