Question: 10. Simplify: $\left(8^{\frac{3}{4}} + 2^3\right)^{\frac{1}{2}}$
(A) $2\sqrt{3}$
(B) $4\sqrt{3}$
(C) $8$
(D) $12$
1. **State the problem:** Simplify the expression $$\left(8^{\frac{3}{4}} + 2^3\right)^{\frac{1}{2}}$$.
2. **Recall the rules:**
- For exponents, $a^{m/n} = \sqrt[n]{a^m}$.
- Simplify inside the parentheses first, then apply the outer exponent.
3. **Simplify each term inside the parentheses:**
- $8^{\frac{3}{4}} = \left(2^3\right)^{\frac{3}{4}} = 2^{3 \times \frac{3}{4}} = 2^{\frac{9}{4}}$
- $2^3 = 8$
4. **Rewrite the expression:**
$$\left(2^{\frac{9}{4}} + 8\right)^{\frac{1}{2}}$$
5. **Express 8 as a power of 2:**
$$8 = 2^3 = 2^{\frac{12}{4}}$$
6. **Add the terms inside the parentheses:**
$$2^{\frac{9}{4}} + 2^{\frac{12}{4}}$$
7. **Factor out the smaller power:**
$$2^{\frac{9}{4}} \left(1 + 2^{\frac{12}{4} - \frac{9}{4}}\right) = 2^{\frac{9}{4}} \left(1 + 2^{\frac{3}{4}}\right)$$
8. **Simplify the expression inside the parentheses:**
$$1 + 2^{\frac{3}{4}}$$
9. **Now the whole expression is:**
$$\left(2^{\frac{9}{4}} \left(1 + 2^{\frac{3}{4}}\right)\right)^{\frac{1}{2}} = \left(2^{\frac{9}{4}}\right)^{\frac{1}{2}} \times \left(1 + 2^{\frac{3}{4}}\right)^{\frac{1}{2}} = 2^{\frac{9}{8}} \times \left(1 + 2^{\frac{3}{4}}\right)^{\frac{1}{2}}$$
10. **Approximate $2^{\frac{3}{4}}$:**
$$2^{\frac{3}{4}} = \sqrt[4]{2^3} = \sqrt[4]{8} \approx 1.68179$$
11. **Calculate inside the parentheses:**
$$1 + 1.68179 = 2.68179$$
12. **Take the square root:**
$$\sqrt{2.68179} \approx 1.637$$
13. **Calculate $2^{\frac{9}{8}}$:**
$$2^{\frac{9}{8}} = 2^{1.125} = 2 \times 2^{0.125} \approx 2 \times 1.0905 = 2.181$$
14. **Multiply the two results:**
$$2.181 \times 1.637 \approx 3.57$$
15. **Compare with answer choices:**
- (A) $2\sqrt{3} \approx 2 \times 1.732 = 3.464$
- (B) $4\sqrt{3} \approx 4 \times 1.732 = 6.928$
- (C) $8$
- (D) $12$
The closest value is (A) $2\sqrt{3}$.
**Final answer:** (A) $2\sqrt{3}$