Evaluate $\frac{27^\frac{1}{3} - 8^\frac{2}{3}}{16^\frac{2}{4} \times 2}$

To solve this expression, we need to simplify each term using the laws of indices. The expression is:

Step 1: Simplify the numerator

• $27^\frac{1}{3}$: This is the cube root of 27. Since $3 \times 3 \times 3 = 27$, then $27^\frac{1}{3} = 3$.
• $8^\frac{2}{3}$: This is $(8^\frac{1}{3})^2$. The cube root of 8 is 2. So, $2^2 = 4$.
• Numerator: $3 - 4 = -1$.

Step 2: Simplify the denominator

• $16^\frac{2}{4}$: First, simplify the fraction in the exponent: $\frac{2}{4} = \frac{1}{2}$. So, $16^\frac{1}{2}$ is the square root of 16, which is 4.
• Denominator: $4 \times 2 = 8$.

Step 3: Combine the results

Now, place the simplified numerator over the simplified denominator:

Analysis

The calculation results in $-\frac{1}{8}$. Comparing this to your options: A. $\frac{23}{5}$ B. $-\frac{23}{5}$ C. $-\frac{1}{8}$ D. $\frac{21}{7}$

The result matches option C.

Exam Tip: When dealing with fractional indices, always try to express the base as a power of a smaller number (e.g., $27 = 3^3$, $8 = 2^3$, $16 = 2^4$). This makes it much easier to cancel out the denominators in the exponents!