Question
Apply the Zero Power Rule to create an equivalent expression to 9⋅(−33)^0+3/2(8−27^0)
Answers
Answer
Apply the Zero Power Rule to create an equivalent expression to 9⋅(−33)^0+3/2(8−27^0) ?
9⋅(−33)^0+3/2(8−27^0) =__ ⋅__ +3/2 (__ – __)
9⋅(−33)^0+3/2(8−27^0) =__ ⋅__ +3/2 (__ – __)
Answer
Solve the expression 15⋅8^0/5+4(7^0−6)
Answer
simplify (10^3)^6
Answer
If the expression 7^15 was simplified using the Power Rule of Exponents, which of the following would be an expanded equivalent expression?(1 point) Responses (7⋅7⋅7⋅7⋅7)^10 (7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7) (7^10)^5 (7^18)^3
Answer
create a simplified equivalent expression for (19^4)^9 .
Answered by
GPT-4o mini
To apply the Zero Power Rule, we recall that any non-zero number raised to the power of zero is equal to 1. Therefore, we can simplify the expression step by step:
1. **Identify terms with zero exponents**:
- \((-33)^0\)
- \(27^0\)
Using the Zero Power Rule:
- \((-33)^0 = 1\)
- \(27^0 = 1\)
2. **Substitute these values back into the expression**:
\[
9 \cdot (−33)^0 + \frac{3}{2}(8 − 27^0)
\]
becomes
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
3. **Simplify each part**:
- The first part: \(9 \cdot 1 = 9\)
- The second part: \(8 - 1 = 7\)
Now substituting these simplified values in:
\[
9 + \frac{3}{2} \cdot 7
\]
4. **Calculate \(\frac{3}{2} \cdot 7\)**:
\[
\frac{3 \cdot 7}{2} = \frac{21}{2}
\]
5. **Combine the results**:
\[
9 + \frac{21}{2}
\]
To combine 9 (as a fraction):
\[
9 = \frac{18}{2}
\]
Now we add the fractions:
\[
\frac{18}{2} + \frac{21}{2} = \frac{18 + 21}{2} = \frac{39}{2}
\]
Thus, the equivalent expression is:
\[
\frac{39}{2}
\]
1. **Identify terms with zero exponents**:
- \((-33)^0\)
- \(27^0\)
Using the Zero Power Rule:
- \((-33)^0 = 1\)
- \(27^0 = 1\)
2. **Substitute these values back into the expression**:
\[
9 \cdot (−33)^0 + \frac{3}{2}(8 − 27^0)
\]
becomes
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
3. **Simplify each part**:
- The first part: \(9 \cdot 1 = 9\)
- The second part: \(8 - 1 = 7\)
Now substituting these simplified values in:
\[
9 + \frac{3}{2} \cdot 7
\]
4. **Calculate \(\frac{3}{2} \cdot 7\)**:
\[
\frac{3 \cdot 7}{2} = \frac{21}{2}
\]
5. **Combine the results**:
\[
9 + \frac{21}{2}
\]
To combine 9 (as a fraction):
\[
9 = \frac{18}{2}
\]
Now we add the fractions:
\[
\frac{18}{2} + \frac{21}{2} = \frac{18 + 21}{2} = \frac{39}{2}
\]
Thus, the equivalent expression is:
\[
\frac{39}{2}
\]
Answered by
GPT-4o mini
Let's apply the Zero Power Rule step by step to express the original expression in the desired format.
Given the expression:
\[
9 \cdot (-33)^0 + \frac{3}{2}(8 - 27^0)
\]
1. **Apply the Zero Power Rule**:
- According to the Zero Power Rule, any non-zero number raised to the power of zero is equal to 1.
- So, \((-33)^0 = 1\) and \(27^0 = 1\).
2. **Substituting these values**:
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
3. **Now write that in the specified format**:
We can rewrite the expression as:
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
This can be expressed in the format \(a \cdot b + \frac{c}{d} (e - f)\):
- Here, we recognize:
- \(a = 9\)
- \(b = 1\)
- \(c = 3\)
- \(d = 2\)
- \(e = 8\)
- \(f = 1\)
Now putting it all together, we have:
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
Thus, the equivalent expression in the format requested is:
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
Given the expression:
\[
9 \cdot (-33)^0 + \frac{3}{2}(8 - 27^0)
\]
1. **Apply the Zero Power Rule**:
- According to the Zero Power Rule, any non-zero number raised to the power of zero is equal to 1.
- So, \((-33)^0 = 1\) and \(27^0 = 1\).
2. **Substituting these values**:
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
3. **Now write that in the specified format**:
We can rewrite the expression as:
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
This can be expressed in the format \(a \cdot b + \frac{c}{d} (e - f)\):
- Here, we recognize:
- \(a = 9\)
- \(b = 1\)
- \(c = 3\)
- \(d = 2\)
- \(e = 8\)
- \(f = 1\)
Now putting it all together, we have:
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
Thus, the equivalent expression in the format requested is:
\[
9 \cdot 1 + \frac{3}{2}(8 - 1)
\]
Answered by
GPT-4o mini
To solve the expression \(15 \cdot 8^0 / 5 + 4(7^0 - 6)\), we will apply the Zero Power Rule. According to the Zero Power Rule, any non-zero number raised to the power of zero is equal to 1.
### Step 1: Evaluate the Zero Exponents
1. \(8^0 = 1\)
2. \(7^0 = 1\)
### Step 2: Substitute the Zero Exponents into the Expression
Now let's substitute these values back into the expression:
\[
15 \cdot 8^0 / 5 + 4(7^0 - 6) = 15 \cdot 1 / 5 + 4(1 - 6)
\]
### Step 3: Simplify Each Term
1. **First term**:
\[
15 \cdot 1 / 5 = 15 / 5 = 3
\]
2. **Second term**:
\[
4(1 - 6) = 4(-5) = -20
\]
### Step 4: Combine the Results
Now, we combine the simplified terms:
\[
3 - 20 = -17
\]
### Final Answer
Thus, the value of the expression is:
\[
\boxed{-17}
\]
### Step 1: Evaluate the Zero Exponents
1. \(8^0 = 1\)
2. \(7^0 = 1\)
### Step 2: Substitute the Zero Exponents into the Expression
Now let's substitute these values back into the expression:
\[
15 \cdot 8^0 / 5 + 4(7^0 - 6) = 15 \cdot 1 / 5 + 4(1 - 6)
\]
### Step 3: Simplify Each Term
1. **First term**:
\[
15 \cdot 1 / 5 = 15 / 5 = 3
\]
2. **Second term**:
\[
4(1 - 6) = 4(-5) = -20
\]
### Step 4: Combine the Results
Now, we combine the simplified terms:
\[
3 - 20 = -17
\]
### Final Answer
Thus, the value of the expression is:
\[
\boxed{-17}
\]
Answered by
GPT-4o mini
To simplify the expression \((10^3)^6\), we use the power of a power property of exponents. According to this property, when you raise a power to another power, you multiply the exponents:
\[
(a^m)^n = a^{m \cdot n}
\]
In this case, we have:
\[
(10^3)^6 = 10^{3 \cdot 6} = 10^{18}
\]
Thus, the simplified expression is:
\[
\boxed{10^{18}}
\]
\[
(a^m)^n = a^{m \cdot n}
\]
In this case, we have:
\[
(10^3)^6 = 10^{3 \cdot 6} = 10^{18}
\]
Thus, the simplified expression is:
\[
\boxed{10^{18}}
\]
Answered by
GPT-4o mini
To simplify and expand \(7^{15}\) using the Power Rule of Exponents, we need to identify which of the given expressions can be simplified to find an equivalent expanded expression.
### Options Analysis
1. **\((7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)^{10}\)**:
- This simplifies to \((7^5)^{10} = 7^{5 \cdot 10} = 7^{50}\).
- This is **not** equivalent to \(7^{15}\).
2. **\((7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7)\)**:
- This expression consists of 5 groups of \(7 \cdot 7 \cdot 7\), each of which is \(7^3\):
- So this becomes \( (7^3)^5 = 7^{3 \cdot 5} = 7^{15}\).
- This is equivalent to \(7^{15}\).
3. **\((7^{10})^5\)**:
- This simplifies to \(7^{10 \cdot 5} = 7^{50}\).
- This is **not** equivalent to \(7^{15}\).
4. **\((7^{18})^3\)**:
- This simplifies to \(7^{18 \cdot 3} = 7^{54}\).
- This is **not** equivalent to \(7^{15}\).
### Conclusion
The only expression that expands to \(7^{15}\) is:
\[
\boxed{(7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7)}
\]
### Options Analysis
1. **\((7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)^{10}\)**:
- This simplifies to \((7^5)^{10} = 7^{5 \cdot 10} = 7^{50}\).
- This is **not** equivalent to \(7^{15}\).
2. **\((7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7)\)**:
- This expression consists of 5 groups of \(7 \cdot 7 \cdot 7\), each of which is \(7^3\):
- So this becomes \( (7^3)^5 = 7^{3 \cdot 5} = 7^{15}\).
- This is equivalent to \(7^{15}\).
3. **\((7^{10})^5\)**:
- This simplifies to \(7^{10 \cdot 5} = 7^{50}\).
- This is **not** equivalent to \(7^{15}\).
4. **\((7^{18})^3\)**:
- This simplifies to \(7^{18 \cdot 3} = 7^{54}\).
- This is **not** equivalent to \(7^{15}\).
### Conclusion
The only expression that expands to \(7^{15}\) is:
\[
\boxed{(7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7)}
\]
Answered by
GPT-4o mini
To simplify the expression \((19^4)^9\), we can use the Power Rule of Exponents, which states that when you raise a power to another power, you multiply the exponents.
The rule can be stated as:
\[
(a^m)^n = a^{m \cdot n}
\]
Applying this to our expression:
\[
(19^4)^9 = 19^{4 \cdot 9}
\]
Now, calculate \(4 \cdot 9\):
\[
4 \cdot 9 = 36
\]
So, we have:
\[
(19^4)^9 = 19^{36}
\]
Thus, the simplified equivalent expression for \((19^4)^9\) is:
\[
\boxed{19^{36}}
\]
The rule can be stated as:
\[
(a^m)^n = a^{m \cdot n}
\]
Applying this to our expression:
\[
(19^4)^9 = 19^{4 \cdot 9}
\]
Now, calculate \(4 \cdot 9\):
\[
4 \cdot 9 = 36
\]
So, we have:
\[
(19^4)^9 = 19^{36}
\]
Thus, the simplified equivalent expression for \((19^4)^9\) is:
\[
\boxed{19^{36}}
\]