in exercises 16–18, rewrite the expression in radical form.
16 ((-3)^{5/3}) 17. (6^{3/2}) 18. (12^{5/4})
in exercises 19–24, evaluate the expression.
19. (32^{2/5}) 20. ((-64)^{3/3}) 21. (343^{2/3})
22. (256^{7/8}) 23. (-729^{5/6}) 24. ((-625)^{3/4})

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Accepted Answer

Let's solve these problems one by one. We'll start with the radical form conversions (Exercises 16 - 18) and then move on to the evaluations (Exercises 19 - 24).

### Exercise 16: Rewrite $(-3)^{5/3}$ in radical form
# Explanation:
## Step 1: Recall the exponent - radical relationship
The formula for converting an expression of the form $a^{m/n}$ to radical form is $a^{m/n}=\sqrt[n]{a^{m}}$ (or equivalently $(\sqrt[n]{a})^{m}$). Here, $a = - 3$, $m = 5$, and $n=3$.
## Step 2: Apply the formula
Using the formula $a^{m/n}=\sqrt[n]{a^{m}}$, we substitute $a=-3$, $m = 5$, and $n = 3$ into it. So $(-3)^{5/3}=\sqrt[3]{(-3)^{5}}$ (we can also write it as $(\sqrt[3]{-3})^{5}$ since $\sqrt[3]{(-3)^{5}}=(\sqrt[3]{-3})^{5}$ because the cube root of a negative number is negative and $(-3)^{5}=-243$ and $\sqrt[3]{-243}=- \sqrt[3]{243}=-3\sqrt[3]{9}$, but the radical form as $\sqrt[3]{(-3)^{5}}$ is also correct).
# Answer: $\sqrt[3]{(-3)^{5}}$ (or $(\sqrt[3]{-3})^{5}$)

### Exercise 17: Rewrite $6^{3/2}$ in radical form
# Explanation:
## Step 1: Recall the exponent - radical relationship
For an expression $a^{m/n}$, the radical form is $\sqrt[n]{a^{m}}$ (or $(\sqrt[n]{a})^{m}$). Here, $a = 6$, $m=3$, and $n = 2$.
## Step 2: Apply the formula
Substitute $a = 6$, $m = 3$, and $n=2$ into the formula $a^{m/n}=\sqrt[n]{a^{m}}$. So $6^{3/2}=\sqrt{6^{3}}$ (or $(\sqrt{6})^{3}$). And $\sqrt{6^{3}}=\sqrt{216} = 6\sqrt{6}$, but the radical form as $\sqrt{6^{3}}$ is acceptable.
# Answer: $\sqrt{6^{3}}$ (or $(\sqrt{6})^{3}$)

### Exercise 18: Rewrite $12^{5/4}$ in radical form
# Explanation:
## Step 1: Recall the exponent - radical relationship
The formula $a^{m/n}=\sqrt[n]{a^{m}}$ (or $(\sqrt[n]{a})^{m}$) holds. Here, $a = 12$, $m = 5$, and $n=4$.
## Step 2: Apply the formula
Substitute $a = 12$, $m = 5$, and $n = 4$ into the formula. So $12^{5/4}=\sqrt[4]{12^{5}}$ (or $(\sqrt[4]{12})^{5}$).
# Answer: $\sqrt[4]{12^{5}}$ (or $(\sqrt[4]{12})^{5}$)

### Exercise 19: Evaluate $32^{2/5}$
# Explanation:
## Step 1: Recall the exponent - radical relationship
We know that $a^{m/n}=\sqrt[n]{a^{m}}=(\sqrt[n]{a})^{m}$. For $32^{2/5}$, we can first find the $n$ - th root of $a$ where $a = 32$ and $n = 5$, then raise it to the power $m = 2$.
## Step 2: Find the 5 - th root of 32
We know that $2^{5}=32$, so $\sqrt[5]{32}=2$.
## Step 3: Raise the result to the power of 2
Now we raise 2 to the power of 2: $2^{2}=4$.
# Answer: 4

### Exercise 20: Evaluate $(-64)^{3/3}$
# Explanation:
## Step 1: Simplify the exponent
First, simplify the exponent $\frac{3}{3}=1$.
## Step 2: Evaluate the expression
So $(-64)^{3/3}=(-64)^{1}=-64$.
# Answer: - 64

### Exercise 21: Evaluate $343^{2/3}$
# Explanation:
## Step 1: Recall the exponent - radical relationship
Using $a^{m/n}=(\sqrt[n]{a})^{m}$, for $a = 343$, $m = 2$, and $n = 3$.
## Step 2: Find the 3 - rd root of 343
We know that $7^{3}=343$, so $\sqrt[3]{343}=7$.
## Step 3: Raise the result to the power of 2
Raise 7 to the power of 2: $7^{2}=49$.
# Answer: 49

### Exercise 22: Evaluate $256^{3/8}$
# Explanation:
## Step 1: Recall the exponent - radical relationship
Using $a^{m/n}=(\sqrt[n]{a})^{m}$, for $a = 256$, $m = 3$, and $n = 8$.
## Step 2: Find the 8 - th root of 256
We know that $2^{8}=256$, so $\sqrt[8]{256}=2$.
## Step 3: Raise the result to the power of 3
Raise 2 to the power of 3: $2^{3}=8$.
# Answer: 8

### Exercise 23: Evaluate $-729^{5/6}$
# Explanation:
## Step 1: Recall the exponent - radical relationship
Using $a^{m/n}=(\sqrt[n]{a})^{m}$, for $a = 729$, $m = 5$, and $n = 6$.
## Step 2: Find the 6 - th root of 729
We know that $3^{6}=729$, so $\sqrt[6]{729}=3$.
## Step 3: Raise the result to the power of 5
Raise 3 to the power of 5: $3^{5}=243$.
## Step 4: Apply the negative sign
Since the original expression is $-729^{5/6}$, the result is $-243$.
# Answer: - 243

### Exercise 24: Evaluate $(-625)^{3/4}$
# Explanation:
## Step 1: Recall the exponent - radical relationship
We know that $a^{m/n}=\sqrt[n]{a^{m}}$. For $(-625)^{3/4}$, we first consider the 4 - th root of $-625$. But the 4 - th root of a negative number is not a real number (since any real number raised to the 4 - th power is non - negative). However, if we consider complex numbers, we can proceed. But if we are working in the real number system, this expression is not defined. If we assume that we can work with the principal root of the absolute value and then consider the sign:
First, $\vert - 625\vert=625$, and $5^{4}=625$, so $\sqrt[4]{625}=5$. But since the base is negative and the index of the root (4) is even, in the real number system, $(-625)^{3/4}$ is not a real number. If we consider the complex number system, we can write $-625 = 625e^{i\pi}$, and $(-625)^{3/4}=(625e^{i\pi})^{3/4}=625^{3/4}e^{i\frac{3\pi}{4}}=(5^{4})^{3/4}(\cos(\frac{3\pi}{4})+i\sin(\frac{3\pi}{4}))=125(-\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})$. But if we are restricted to real numbers, the expression is undefined.

If we made a mistake in the problem (maybe it's $(-625)^{3/5}$ instead of $(-625)^{3/4}$), let's check: For $(-625)^{3/5}$, $\sqrt[5]{-625}=-5$ (since $(-5)^{5}=-625$), and then $(-5)^{3}=-125$. But as per the given problem, it's $(-625)^{3/4}$. In the real number system, the 4 - th root of a negative number is not real, so the expression $(-625)^{3/4}$ is not a real number. If we assume that the problem has a typo and the exponent is $3/5$, the answer would be $-125$. But based on the given problem, in the real number system, it's undefined.

If we consider the principal root in the complex plane, the answer is $125(-\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})\approx - 88.388 + 88.388i$. But if we are working in real numbers, we can say that the expression is not defined.

(If we assume that the problem meant $(625)^{3/4}$, then $\sqrt[4]{625}=5$, $5^{3}=125$)

Since the problem is likely expecting a real - valued answer, there might be a typo. But if we proceed with the given problem:

In the real number system, $(-625)^{3/4}$ is not defined. If we consider the complex number system, the answer is $125\sqrt{2}(-\frac{1}{2}+i\frac{1}{2})$ or $ - 62.5\sqrt{2}+62.5\sqrt{2}i\approx - 88.39 + 88.39i$

# Answer: In real numbers, undefined; In complex numbers, $125(-\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})$ (or approximately $-88.39 + 88.39i$)

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QUESTION IMAGE

Question

Explanation:

Exercise 16: Rewrite \((-3)^{5/3}\) in radical form

Step 1: Recall the exponent - radical relationship

Step 2: Apply the formula

Step 1: Recall the exponent - radical relationship

Step 2: Apply the formula

Step 1: Recall the exponent - radical relationship

Step 2: Apply the formula

Answer:

Exercise 17: Rewrite \(6^{3/2}\) in radical form