Question

which of the following statements are true? select all that apply. assume that we are restricting all values and results to positive numbers.
□ $\left(\sqrt5{7}\
ight)^5 = 7$
□ $\left(\sqrt5{7}\
ight)\left(\sqrt5{7}\
ight) = 7$
□ $\left(\sqrt5{7}\
ight)^8 = 7$
□ $\left(\sqrt5{7}\
ight)^7 = 7$
□ $\left(\sqrt{7}\
ight)\left(\sqrt{7}\
ight) = 7$
□ $\left(\sqrt5{7}\
ight)\left(\sqrt5{7}\
ight)\left(\sqrt5{7}\
ight) = 7$
□ $\left(\sqrt5{7}\
ight)\left(\sqrt5{7}\
ight) = 7$
points possible : 2
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Explanation:

Step1: Recall exponent rules

For a positive real number \( a \) and rational exponents, \( \sqrt[n]{a}=a^{\frac{1}{n}} \) and \( (a^m)(a^n)=a^{m + n} \), \( (a^m)^n=a^{mn} \).

Step2: Analyze \( (\sqrt[5]{7})^5 \)

Using \( (a^m)^n=a^{mn} \), here \( a = 7 \), \( m=\frac{1}{5} \), \( n = 5 \). So \( (\sqrt[5]{7})^5=(7^{\frac{1}{5}})^5=7^{\frac{1}{5}\times5}=7^1 = 7 \). This is true.

Step3: Analyze \( (\sqrt[5]{7})(\sqrt[5]{7}) \)

Using \( (a^m)(a^n)=a^{m + n} \), \( \sqrt[5]{7}=7^{\frac{1}{5}} \), so \( (7^{\frac{1}{5}})(7^{\frac{1}{5}})=7^{\frac{1}{5}+\frac{1}{5}}=7^{\frac{2}{5}} eq7 \). Wait, no, wait the third option? Wait, no, let's check each:

Wait, second option: \( (\sqrt[5]{7})(\sqrt[5]{7}) \): \( 7^{\frac{1}{5}}\times7^{\frac{1}{5}}=7^{\frac{2}{5}} eq7 \). Wait, third option: \( (\sqrt{7})(\sqrt{7}) \): \( \sqrt{7}=7^{\frac{1}{2}} \), so \( 7^{\frac{1}{2}}\times7^{\frac{1}{2}}=7^{\frac{1}{2}+\frac{1}{2}}=7^1 = 7 \). This is true.

Fourth option: \( (\sqrt[5]{7})^7=7^{\frac{7}{5}} eq7 \).

Fifth option: \( (\sqrt[5]{7})^8=7^{\frac{8}{5}} eq7 \).

Sixth option: \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})=7^{\frac{1}{5}+\frac{1}{5}+\frac{1}{5}}=7^{\frac{3}{5}} eq7 \).

Seventh option: \( (\sqrt[5]{7})(\sqrt[5]{7})=7^{\frac{2}{5}} eq7 \)? Wait no, wait the first option: \( (\sqrt[5]{7})^5 = 7 \) (true), third option: \( (\sqrt{7})(\sqrt{7})=7 \) (true), and let's re - check the seventh? Wait no, the options are:

1. \( (\sqrt[5]{7})^5 = 7 \)
2. \( (\sqrt[5]{7})(\sqrt[5]{7})=7 \)
3. \( (\sqrt{7})(\sqrt{7})=7 \)
4. \( (\sqrt[5]{7})^7 = 7 \)
5. \( (\sqrt[5]{7})^8 = 7 \)
6. \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})=7 \)
7. \( (\sqrt[5]{7})(\sqrt[5]{7})=7 \) (Wait, maybe typo, but let's use exponent rules correctly)

Wait, \( (\sqrt[5]{7})^5=7^{\frac{1}{5}\times5}=7 \) (true).

\( (\sqrt{7})(\sqrt{7})=(7^{\frac{1}{2}})(7^{\frac{1}{2}})=7^{\frac{1}{2}+\frac{1}{2}}=7 \) (true).

Also, let's check the last option: \( (\sqrt[5]{7})(\sqrt[5]{7}) \): no, wait the seventh option is \( (\sqrt[5]{7})(\sqrt[5]{7}) \), which is \( 7^{\frac{2}{5}} eq7 \). Wait, maybe I misread. Wait the options are:

- \( (\sqrt[5]{7})^5 = 7 \)
- \( (\sqrt[5]{7})(\sqrt[5]{7})=7 \)
- \( (\sqrt{7})(\sqrt{7})=7 \)
- \( (\sqrt[5]{7})^7 = 7 \)
- \( (\sqrt[5]{7})^8 = 7 \)
- \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})=7 \)
- \( (\sqrt[5]{7})(\sqrt[5]{7})=7 \)

Wait, no, the first option: \( (\sqrt[5]{7})^5 = 7 \) (true, by \( (a^{\frac{1}{n}})^n=a \)).

Third option: \( (\sqrt{7})(\sqrt{7})=7 \) (true, by \( (a^{\frac{1}{2}})(a^{\frac{1}{2}})=a \)).

Also, let's check the sixth option: \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7}) \): no, that's \( 7^{\frac{3}{5}} eq7 \). Wait, maybe the second option is a typo, but according to exponent rules:

- \( (\sqrt[5]{7})^5 = 7 \): True.
- \( (\sqrt{7})(\sqrt{7})=7 \): True.
- Let's check the seventh option: \( (\sqrt[5]{7})(\sqrt[5]{7}) \): no, that's \( 7^{\frac{2}{5}} eq7 \). Wait, maybe I made a mistake. Wait the problem says "select all that apply".

Wait, \( (\sqrt[5]{7})^5 = 7 \): true.

\( (\sqrt{7})(\sqrt{7})=7 \): true (since \( \sqrt{7}\times\sqrt{7}=7 \)).

Also, let's check \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})=(\sqrt[5]{7})^5 = 7 \), but the options:

Wait the first option is \( (\sqrt[5]{7})^5 = 7 \) (true).

The third option: \( (\sqrt{7})(\sqrt{7})=7 \) (true).

Is there another? Wait the seventh option: \( (\sqrt[5]{7})(\sqrt[5]{7}) \): no. Wait maybe the second option is \( (\sqrt[5]{7})^5 \)? No, the first option is \( (\sqrt[5]{7})^5 = 7 \).

Wait, let's re - express each:

1. \( (\sqrt[5]{7})^5=7^{\frac{1}{5}\times5}=7^1 = 7 \): True.

2. \( (\sqrt[5]{7})(\sqrt[5]{7})=7^{\frac{1}{5}+\frac{1}{5}}=7^{\frac{2}{5}} eq7 \): False.

3. \( (\sqrt{7})(\sqrt{7})=7^{\frac{1}{2}+\frac{1}{2}}=7^1 = 7 \): True.

4. \( (\sqrt[5]{7})^7=7^{\frac{7}{5}} eq7 \): False.

5. \( (\sqrt[5]{7})^8=7^{\frac{8}{5}} eq7 \): False.

6. \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})=7^{\frac{3}{5}} eq7 \): False.

7. \( (\sqrt[5]{7})(\sqrt[5]{7})=7^{\frac{2}{5}} eq7 \): False.

So the true ones are \( (\sqrt[5]{7})^5 = 7 \) and \( (\sqrt{7})(\sqrt{7})=7 \), and also, wait the first option and the third option, and is there another? Wait, wait the seventh option: no, maybe I misread the options. Wait the user's image:

Options:

- \( (\sqrt[5]{7})^5 = 7 \)

- \( (\sqrt[5]{7})(\sqrt[5]{7})=7 \)

- \( (\sqrt{7})(\sqrt{7})=7 \)

- \( (\sqrt[5]{7})^7 = 7 \)

- \( (\sqrt[5]{7})^8 = 7 \)

- \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})=7 \)

- \( (\sqrt[5]{7})(\sqrt[5]{7})=7 \)

Wait, maybe the sixth option is \( (\sqrt[5]{7})^3 \)? No, but according to exponent rules, only \( (\sqrt[5]{7})^5 = 7 \) and \( (\sqrt{7})(\sqrt{7})=7 \) and also, wait \( (\sqrt[5]{7})^5 = 7 \), \( (\sqrt{7})(\sqrt{7})=7 \), and also, let's check the first option and the third option, and is there a mistake in my analysis?

Wait, \( (\sqrt[5]{7})^5 = 7 \): true.

\( (\sqrt{7})(\sqrt{7})=7 \): true.

Also, \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7})=(\sqrt[5]{7})^5 = 7 \), but that's the first option.

Wait, maybe the seventh option is a typo, but based on exponent rules, the true statements are \( (\sqrt[5]{7})^5 = 7 \) and \( (\sqrt{7})(\sqrt{7})=7 \), and also, wait the first option and the third option, and is there another? Wait, no, let's confirm:

For \( (\sqrt[5]{7})^5 \): By definition of the fifth root, raising the fifth root of 7 to the fifth power gives 7. So this is true.

For \( (\sqrt{7})(\sqrt{7}) \): The square root of 7 times the square root of 7 is 7, since \( \sqrt{a}\times\sqrt{a}=a \) for \( a\geq0 \). So this is true.

For \( (\sqrt[5]{7})(\sqrt[5]{7}) \): This is \( 7^{\frac{1}{5}+\frac{1}{5}}=7^{\frac{2}{5}} eq7 \), so false.

For \( (\sqrt[5]{7})^7 \): \( 7^{\frac{7}{5}} eq7 \), false.

For \( (\sqrt[5]{7})^8 \): \( 7^{\frac{8}{5}} eq7 \), false.

For \( (\sqrt[5]{7})(\sqrt[5]{7})(\sqrt[5]{7}) \): \( 7^{\frac{3}{5}} eq7 \), false.

For the last option \( (\sqrt[5]{7})(\sqrt[5]{7}) \): same as the second option, false.

So the true statements are \( (\sqrt[5]{7})^5 = 7 \) and \( (\sqrt{7})(\sqrt{7})=7 \).

Answer:

\( (\sqrt[5]{7})^5 = 7 \), \( (\sqrt{7})(\sqrt{7})=7 \) (the first and the third options in the list)