Simplify $(\frac{16}{81})^{\frac{1}{4}} \div (\frac{9}{16})^{-\frac{1}{2}}$

To solve this expression, we need to apply the laws of indices, specifically the rules for fractional exponents and negative exponents.

1. Analyze the expression

The expression is: $(\frac{16}{81})^{\frac{1}{4}} \div (\frac{9}{16})^{-\frac{1}{2}}$

2. Explain the concepts

• Fractional Exponents: $x^{\frac{1}{n}}$ is the same as the $n^{th}$ root of $x$. For example, $x^{\frac{1}{4}}$ means $\sqrt[4]{x}$.
• Negative Exponents: $x^{-n} = (\frac{1}{x})^n$. This means if you have a fraction raised to a negative power, you flip the fraction (find the reciprocal) to make the exponent positive: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.

3. Step-by-Step Solution

Step 1: Simplify the first part $(\frac{16}{81})^{\frac{1}{4}}$ We look for a number that, when raised to the power of 4, gives 16 and 81.

• $16 = 2^4$
• $81 = 3^4$ So, $(\frac{2^4}{3^4})^{\frac{1}{4}} = (\frac{2}{3})^{4 \times \frac{1}{4}} = (\frac{2}{3})^1 = \mathbf{\frac{2}{3}}$

Step 2: Simplify the second part $(\frac{9}{16})^{-\frac{1}{2}}$ First, remove the negative sign by flipping the fraction: $(\frac{9}{16})^{-\frac{1}{2}} = (\frac{16}{9})^{\frac{1}{2}}$ Now, calculate the square root ($\frac{1}{2}$ power) of the fraction:

• $\sqrt{16} = 4$
• $\sqrt{9} = 3$ So, $(\frac{16}{9})^{\frac{1}{2}} = \mathbf{\frac{4}{3}}$

Step 3: Perform the division Now substitute the simplified values back into the original expression: $\frac{2}{3} \div \frac{4}{3}$ To divide by a fraction, multiply by its reciprocal: $\frac{2}{3} \times \frac{3}{4}$ The 3s cancel out: $\frac{2}{4} = \mathbf{\frac{1}{2}}$

4. Identify the answer

Comparing our result to the given options, we find that the correct choice is B.

5. Exam Tip

When you see a negative exponent on a fraction, always flip the fraction first to make the exponent positive before doing any other calculation. It prevents errors with signs and makes the numbers much easier to work with!