4.3 Calculate the following without the use of a calculator: $$ \begin{array}{l}4.3 .1 \ -2+(-4)(-2)-3 \ 4.3 .2 \ 8+9 imes 10 \div(3+7) \ 4.3 .3\end{array} 3^{3}-11^{2}+\sqrt[4]{-256}-(-5)^{3} $$

Question

4.3 Calculate the following without the use of a calculator: $$ \begin{array}{l}4.3 .1 \ -2+(-4)(-2)-3 \ 4.3 .2 \ 8+9 	imes 10 \div(3+7) \ 4.3 .3\end{array} 3^{3}-11^{2}+\sqrt[4]{-256}-(-5)^{3} $$

UpStudy AI · Accepted Answer

**4.3.1**

We need to compute
$$
-2 + (-4)(-2) - 3.
$$
1. First, multiply $$-4$$ and $$-2$$:
   $$
   (-4)(-2) = 8.
   $$
2. Substitute back into the expression:
   $$
   -2 + 8 - 3.
   $$
3. Combine step by step:
   $$
   -2 + 8 = 6,
   $$
   $$
   6 - 3 = 3.
   $$

Thus, the answer is $$\boxed{3}$$.

---

**4.3.2**

We need to compute
$$
8 + 9 \times 10 \div (3+7).
$$
1. Evaluate the parentheses:
   $$
   3+7 = 10.
   $$
2. Substitute back:
   $$
   8 + 9 \times 10 \div 10.
   $$
3. Perform the multiplication and division from left to right:
   $$
   9 \times 10 = 90,
   $$
   $$
   90 \div 10 = 9.
   $$
4. Finally, add:
   $$
   8 + 9 = 17.
   $$

Thus, the answer is $$\boxed{17}$$.

---

**4.3.3**

We need to compute
$$
3^{3} - 11^{2} + \sqrt[4]{-256} - (-5)^{3}.
$$

*Note:* The expression $$\sqrt[4]{-256}$$ involves taking an even-index root of a negative number. In the real number system this is not defined; however, assuming we work with complex numbers, we can find its principal fourth root.

1. Calculate $$3^3$$:
   $$
   3^3 = 27.
   $$
2. Calculate $$11^2$$:
   $$
   11^2 = 121.
   $$
3. Find the principal fourth root of $$-256$$:

Write $$-256$$ in polar form as
   $$
   -256 = 256\bigl(\cos \pi + i\sin \pi\bigr).
   $$
   The fourth root is given by
   $$
   \sqrt[4]{-256} = 256^{\frac{1}{4}} \left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right).
   $$
   Since
   $$
   256^{\frac{1}{4}} = 4,
   $$
   we have
   $$
   \sqrt[4]{-256} = 4\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) = 4\left(\frac{\sqrt{2}}{2}+ i\frac{\sqrt{2}}{2}\right) = 2\sqrt{2} + 2\sqrt{2}i.
   $$
4. Calculate $$(-5)^3$$:
   $$
   (-5)^3 = -125,
   $$
   so
   $$
   -(-5)^3 = -(-125) = 125.
   $$
5. Now substitute all parts into the expression:
   $$
   27 - 121 + \Bigl(2\sqrt{2}+2\sqrt{2}i\Bigr) + 125.
   $$
6. Combine the real parts:
   $$
   27 - 121 + 125 = 31.
   $$
7. Therefore, the complete expression becomes:
   $$
   31 + 2\sqrt{2} + 2\sqrt{2}i.
   $$

Thus, if we allow complex numbers, the answer is 
$$
\boxed{31 + 2\sqrt{2} + 2\sqrt{2}i}.
$$
**4.3.1** $$ -2 + (-4)(-2) - 3 = 3 $$ **4.3.2** $$ 8 + 9 \times 10 \div (3+7) = 17 $$ **4.3.3** $$ 3^{3} - 11^{2} + \sqrt[4]{-256} - (-5)^{3} = 31 + 2\sqrt{2} + 2\sqrt{2}i $$

4.3 Calculate the following without the use of a calculator: \( \begin{array}{l}4.3 .1 \\ -2+(-4)(-2)-3 \\ 4.3 .2 \\ 8+9 \times 10 \div(3+7) \\ 4.3 .3\end{array} 3^{3}-11^{2}+\sqrt[4]{-256}-(-5)^{3} \)

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