Step 1: Understanding the Expression
To solve the expression (a^1/3 × a^1/2 × a^1/4) ÷ a^2/3, we first need to simplify the numerator.
Step 2: Simplifying the Numerator
- We combine the exponents of 'a' in the numerator using the property of exponents: a^m × a^n = a^(m+n).
- Here, we have:
- 1/3 + 1/2 + 1/4.
- To add these fractions, we need a common denominator. The least common multiple of 3, 2, and 4 is 12.
- Converting each fraction:
- 1/3 = 4/12,
- 1/2 = 6/12,
- 1/4 = 3/12.
- Now we sum them:
- 4/12 + 6/12 + 3/12 = 13/12.
This means the numerator simplifies to a^(13/12).
Step 3: Dividing by the Denominator
- We rewrite the expression now as:
- (a^(13/12)) ÷ (a^(2/3)).
- Again, we apply the exponent rule: a^m ÷ a^n = a^(m-n).
- First, we convert 2/3 to have a common denominator with 13/12:
- 2/3 = 8/12.
- Now, we perform the subtraction:
- 13/12 - 8/12 = 5/12.
Step 4: Final Result
Thus, the final value of the expression is:
- a^(5/12).
This provides a clear and simplified answer to the original expression.