Evaluate ($\frac{1}{343}$)$^{\frac{1}{3}}$ + (64)$^{-\frac{1}{3}}$ - $\frac{4}{9}$$^{-\frac{1}{2}}$

Analyze the question: We have three terms with fractional and negative indices. We need to evaluate each one separately and then combine them.

Concept:

• a^(1/3) means the cube root of a.
• a^(-n) = 1/(a^n), and (a/b)^(-n) = (b/a)^n.
• a^(-1/2) means the reciprocal of the square root of a.

Solution:

Term 1: (1/343)^(1/3) = 1/(cube root of 343) = 1/7

Term 2: 64^(-1/3) = 1/(cube root of 64) = 1/4

Term 3: (4/9)^(-1/2) = (9/4)^(1/2) = 3/2

(Reading the last term as (4/9)^(-1/2), which is the standard reading.)

Combine:

1/7 + 1/4 - 3/2

LCD = 28:

4/28 + 7/28 - 42/28 = (4 + 7 - 42)/28 = -31/28

Answer: -31/28, which is option C.

Tip: For a negative index on a fraction, flip the fraction and make the index positive. Then take the root first, because it keeps the numbers small. Also, always check whether the exponent applies to the whole bracket, as in (4/9)^(-1/2), or just to part of it.