Question

Whcih of the following is true?
A. \(\left(-11\right)^{0}=1\)
B. \(\frac{11^{3}}{11^{3}}=11^{1}\)
C. \(-11^{0}=1\)
D. \(11^{5}\div 11^{4}=1\)
2. Apply the power of zero rule of exponets \(\left(a^{4}\right)\ \left(a^{0}\right)\ \left(a^{5}\right)\)
3. What is the value of (\(\frac{2}{3}^{0}\)/)

Answer 1

Let's analyze each statement one by one:

A. \(\left(-11\right)^{0}=1\)

This statement is **true**. Any non-zero number raised to the power of zero is 1, and \(-11\) is a non-zero number.

B. \(\frac{11^{3}}{11^{3}}=11^{1}\)

This statement is **false**. According to the properties of exponents, \(\frac{a^m}{a^n} = a^{m-n}\), therefore:
\[
\frac{11^{3}}{11^{3}} = 11^{3-3} = 11^{0} = 1.
\]

C. \(-11^{0}=1\)

This statement is **false** because \(-11^{0}\) is interpreted as \(-(11^{0})\). Since \(11^{0} = 1\), it would be \(-1\) instead of \(1\).

D. \(11^{5}\div 11^{4}=1\)

This statement is **false**. According to the properties of exponents:
\[
11^{5}\div 11^{4} = 11^{5-4} = 11^{1} = 11.
\]

Summary:
- A is true.
- B is false.
- C is false.
- D is false.

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2. Apply the power of zero rule of exponents: \((a^{4}) \cdot (a^{0}) \cdot (a^{5})\)

Using the power of zero rule, \(a^{0} = 1\), we have:
\[
(a^{4}) \cdot (a^{0}) \cdot (a^{5}) = (a^{4}) \cdot (1) \cdot (a^{5}) = a^{4} \cdot a^{5}.
\]
Now, using the property of exponents:
\[
a^{4} \cdot a^{5} = a^{4+5} = a^{9}.
\]

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3. What is the value of \(\left(\frac{2}{3}\right)^{0}\)?

According to the power of zero rule, any non-zero number raised to the power of zero is 1. Therefore:
\[
\left(\frac{2}{3}\right)^{0} = 1.
\]

In summary, the results are:
1. A is true; B, C, and D are false.
2. The result of \((a^{4}) \cdot (a^{0}) \cdot (a^{5})\) is \(a^{9}\).
3. \(\left(\frac{2}{3}\right)^{0} = 1\).