1. **State the problem:** Write each expression as a single power of 4. 2. **Recall:** - $\sqrt{a} = a^{1/2}$ - $\sqrt[n]{a} = a^{1/n}$ 3. **Solve each part:** **a) $(\sqrt{16})^3$** - First, rewrite $\sqrt{16} = 16^{1/2}$. - Since $16 = 4^2$, then $16^{1/2} = (4^2)^{1/2} = 4^{2 \times \frac{1}{2}} = 4^1 = 4$. - Now raise it to the power 3: $(4)^3 = 4^3$. - **Answer:** $4^3$. **b) $\sqrt[3]{16}$** - Rewrite as $16^{1/3}$. - Substitute $16 = 4^2$ so $16^{1/3} = (4^2)^{1/3} = 4^{2/3}$. - **Answer:** $4^{2/3}$. **c) $\sqrt{64} \times (\sqrt[4]{128})^3$** - Rewrite $\sqrt{64} = 64^{1/2}$ and $\sqrt[4]{128} = 128^{1/4}$. - Express $64$ and $128$ as powers of 4: - $64 = 4^3$ (since $4^3 = 64$) - $128 = 2^7 = (4^{1/2})^7 = 4^{7/2}$ - So, - $64^{1/2} = (4^3)^{1/2} = 4^{3/2}$ - $(128^{1/4})^3 = (4^{7/2})^{1/4 \times 3} = 4^{\frac{7}{2} \times \frac{3}{4}} = 4^{21/8}$ - Multiply powers of 4: $4^{3/2} \times 4^{21/8} = 4^{\frac{3}{2} + \frac{21}{8}} = 4^{\frac{12}{8} + \frac{21}{8}} = 4^{33/8}$. - **Answer:** $4^{33/8}$. **d) $(\sqrt{2})^8 \times (\sqrt[3]{2})^4$** - Rewrite: - $\sqrt{2} = 2^{1/2}$ so $(\sqrt{2})^8 = (2^{1/2})^8 = 2^{8/2} = 2^4$ - $\sqrt[3]{2} = 2^{1/3}$ so $(\sqrt[3]{2})^4 = (2^{1/3})^4 = 2^{4/3}$ - Multiply powers of 2: $2^4 \times 2^{4/3} = 2^{4 + 4/3} = 2^{\frac{12}{3} + \frac{4}{3}} = 2^{16/3}$ - Now express base 2 as power of 4: since $2 = 4^{1/2}$, $$2^{16/3} = (4^{1/2})^{16/3} = 4^{\frac{1}{2} \times \frac{16}{3}} = 4^{8/3}$$ - **Answer:** $4^{8/3}$. **Final answers:** **a)** $4^3$ **b)** $4^{2/3}$ **c)** $4^{33/8}$ **d)** $4^{8/3}$