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Apply the Properties of Integer Exponents to generate equivalent expressions to 3^7 times 3^-9 with only positive exponents, th...Question
Apply the Properties of Integer Exponents to generate equivalent expressions to 3^7 times 3^-9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
Pls explain simply
Pls explain simply
Answers
Answer
What's /[
\frac{1}{9}
\]???
Idk what it means
\frac{1}{9}
\]???
Idk what it means
Answer
Does frac stand for fraction and what are the 1 and 9, are they exponents?
Answer
Ok that was correct (1 over 9)
Now, apply the property of integer exponents to generate equivalent expressions to 2^2 over 2^-4 with only positive exponents. solve the expression and provide the solution as a whole number
Now, apply the property of integer exponents to generate equivalent expressions to 2^2 over 2^-4 with only positive exponents. solve the expression and provide the solution as a whole number
Answer
Correct.
Next, apply the properties of integer exponentd to 4^0 times 5^-2 over (-2)^-3 with only positive exponents. Your answer will be a simplified fraction with no remaining exponents
Next, apply the properties of integer exponentd to 4^0 times 5^-2 over (-2)^-3 with only positive exponents. Your answer will be a simplified fraction with no remaining exponents
Answer
idk if I really understood that bc I got it wrong but...
Apply the property of integer exponents to generate the equivalent expressions to (14^3 times 27^-7)^0 with only positive exponents then solve the expression. your answer will be a simplified fraction with no remaining exponent
Apply the property of integer exponents to generate the equivalent expressions to (14^3 times 27^-7)^0 with only positive exponents then solve the expression. your answer will be a simplified fraction with no remaining exponent
Answered by
GPT-4o mini
To begin solving the expression \(3^7 \times 3^{-9}\) using the properties of integer exponents, we can use the property that states:
\[
a^m \times a^n = a^{m+n}
\]
In this case, we let \(a = 3\), \(m = 7\), and \(n = -9\):
\[
3^7 \times 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2}
\]
Now, since we want to write this with only positive exponents, we can apply the negative exponent rule, which states:
\[
a^{-n} = \frac{1}{a^n}
\]
Using this on \(3^{-2}\):
\[
3^{-2} = \frac{1}{3^2}
\]
Next, we compute \(3^2\):
\[
3^2 = 9
\]
Thus, we can rewrite our expression as:
\[
3^{-2} = \frac{1}{9}
\]
So the equivalent expression of \(3^7 \times 3^{-9}\) with only positive exponents is \(\frac{1}{9}\).
Therefore, the final answer is:
\[
\frac{1}{9}
\]
\[
a^m \times a^n = a^{m+n}
\]
In this case, we let \(a = 3\), \(m = 7\), and \(n = -9\):
\[
3^7 \times 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2}
\]
Now, since we want to write this with only positive exponents, we can apply the negative exponent rule, which states:
\[
a^{-n} = \frac{1}{a^n}
\]
Using this on \(3^{-2}\):
\[
3^{-2} = \frac{1}{3^2}
\]
Next, we compute \(3^2\):
\[
3^2 = 9
\]
Thus, we can rewrite our expression as:
\[
3^{-2} = \frac{1}{9}
\]
So the equivalent expression of \(3^7 \times 3^{-9}\) with only positive exponents is \(\frac{1}{9}\).
Therefore, the final answer is:
\[
\frac{1}{9}
\]
Answered by
GPT-4o mini
I apologize for the confusion! It looks like there was an unintended formatting issue in my response.
The expression \(\frac{1}{9}\) is simply a mathematical way to express "one ninth." It means you have a fraction where 1 is the numerator (the top number) and 9 is the denominator (the bottom number).
To clarify, the final answer to your original question is:
\[
\frac{1}{9}
\]
This means that after applying the properties of integer exponents to simplify \(3^7 \times 3^{-9}\), we find that the result is \(\frac{1}{9}\). If you have any more questions or need further clarification, feel free to ask!
The expression \(\frac{1}{9}\) is simply a mathematical way to express "one ninth." It means you have a fraction where 1 is the numerator (the top number) and 9 is the denominator (the bottom number).
To clarify, the final answer to your original question is:
\[
\frac{1}{9}
\]
This means that after applying the properties of integer exponents to simplify \(3^7 \times 3^{-9}\), we find that the result is \(\frac{1}{9}\). If you have any more questions or need further clarification, feel free to ask!
Answered by
GPT-4o mini
Yes, "frac" is short for "fraction," and in mathematical notation, it is used to represent a fraction.
In the expression \(\frac{1}{9}\):
- The **1** is the **numerator** (the top part of the fraction).
- The **9** is the **denominator** (the bottom part of the fraction).
Neither the **1** nor the **9** are exponents; they are simply the numbers that make up the fraction. This means that \(\frac{1}{9}\) represents one part out of nine equal parts.
So, to clarify the components:
- **\(\frac{1}{9}\)** means "one divided by nine," which is a simple fraction.
- There are no exponents involved in this expression.
If you need more information or have other questions, just let me know!
In the expression \(\frac{1}{9}\):
- The **1** is the **numerator** (the top part of the fraction).
- The **9** is the **denominator** (the bottom part of the fraction).
Neither the **1** nor the **9** are exponents; they are simply the numbers that make up the fraction. This means that \(\frac{1}{9}\) represents one part out of nine equal parts.
So, to clarify the components:
- **\(\frac{1}{9}\)** means "one divided by nine," which is a simple fraction.
- There are no exponents involved in this expression.
If you need more information or have other questions, just let me know!
Answered by
GPT-4o mini
To solve the expression \(\frac{2^2}{2^{-4}}\) using the properties of integer exponents, we can use the property that states:
\[
\frac{a^m}{a^n} = a^{m-n}
\]
In this case, we have:
- \(a = 2\)
- \(m = 2\)
- \(n = -4\)
Now applying the property:
\[
\frac{2^2}{2^{-4}} = 2^{2 - (-4)} = 2^{2 + 4} = 2^6
\]
Now, we compute \(2^6\):
\[
2^6 = 64
\]
Thus, the expression \(\frac{2^2}{2^{-4}}\) simplifies to \(64\).
The final answer is:
\[
64
\]
\[
\frac{a^m}{a^n} = a^{m-n}
\]
In this case, we have:
- \(a = 2\)
- \(m = 2\)
- \(n = -4\)
Now applying the property:
\[
\frac{2^2}{2^{-4}} = 2^{2 - (-4)} = 2^{2 + 4} = 2^6
\]
Now, we compute \(2^6\):
\[
2^6 = 64
\]
Thus, the expression \(\frac{2^2}{2^{-4}}\) simplifies to \(64\).
The final answer is:
\[
64
\]
Answered by
GPT-4o mini
To simplify the expression \(\frac{4^0 \times 5^{-2}}{(-2)^{-3}}\) using the properties of integer exponents, we will break it down step by step.
1. **Evaluating \(4^0\)**:
- According to the property of exponents, any non-zero number raised to the power of 0 is equal to 1. So, we have:
\[
4^0 = 1
\]
2. **Substituting 4^0 into the expression**:
- This allows us to simplify our original expression:
\[
\frac{1 \times 5^{-2}}{(-2)^{-3}} = \frac{5^{-2}}{(-2)^{-3}}
\]
3. **Using the property of negative exponents**:
- Recall that \(a^{-n} = \frac{1}{a^n}\). This allows us to rewrite the expression:
\[
5^{-2} = \frac{1}{5^2}
\]
\[
(-2)^{-3} = \frac{1}{(-2)^3}
\]
Using these, we can update our expression:
\[
\frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{5^2}}{\frac{1}{(-2)^3}}
\]
4. **Dividing fractions**:
- Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[
\frac{1}{5^2} \times \frac{(-2)^3}{1} = \frac{(-2)^3}{5^2}
\]
5. **Calculating the values**:
- Now, we calculate \((-2)^3\) and \(5^2\):
\[
(-2)^3 = -8
\]
\[
5^2 = 25
\]
6. **Combining these values**:
- Now we can combine them into our fraction:
\[
\frac{-8}{25}
\]
Since we want the expression with only positive exponents and denominators, we can rewrite this as:
\[
-\frac{8}{25}
\]
However, if we're focusing on just the magnitude and ignoring the negative sign for the simplified fraction, the answer remains \(-\frac{8}{25}\). If you were looking for just the positive component in terms of its structure, the expression simplifies as shown.
Thus, the final answer is:
\[
-\frac{8}{25}
\]
1. **Evaluating \(4^0\)**:
- According to the property of exponents, any non-zero number raised to the power of 0 is equal to 1. So, we have:
\[
4^0 = 1
\]
2. **Substituting 4^0 into the expression**:
- This allows us to simplify our original expression:
\[
\frac{1 \times 5^{-2}}{(-2)^{-3}} = \frac{5^{-2}}{(-2)^{-3}}
\]
3. **Using the property of negative exponents**:
- Recall that \(a^{-n} = \frac{1}{a^n}\). This allows us to rewrite the expression:
\[
5^{-2} = \frac{1}{5^2}
\]
\[
(-2)^{-3} = \frac{1}{(-2)^3}
\]
Using these, we can update our expression:
\[
\frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{5^2}}{\frac{1}{(-2)^3}}
\]
4. **Dividing fractions**:
- Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[
\frac{1}{5^2} \times \frac{(-2)^3}{1} = \frac{(-2)^3}{5^2}
\]
5. **Calculating the values**:
- Now, we calculate \((-2)^3\) and \(5^2\):
\[
(-2)^3 = -8
\]
\[
5^2 = 25
\]
6. **Combining these values**:
- Now we can combine them into our fraction:
\[
\frac{-8}{25}
\]
Since we want the expression with only positive exponents and denominators, we can rewrite this as:
\[
-\frac{8}{25}
\]
However, if we're focusing on just the magnitude and ignoring the negative sign for the simplified fraction, the answer remains \(-\frac{8}{25}\). If you were looking for just the positive component in terms of its structure, the expression simplifies as shown.
Thus, the final answer is:
\[
-\frac{8}{25}
\]
Answered by
GPT-4o mini
I’m glad you’re continuing with your practice! Let's break down the expression \((14^3 \times 27^{-7})^0\) step by step using the properties of integer exponents.
1. **Using the Zero Exponent Rule**:
- According to the property of exponents, any non-zero number raised to the power of 0 is equal to 1. Thus, we have:
\[
(14^3 \times 27^{-7})^0 = 1
\]
2. **Expressing with Positive Exponents**:
- Since \(1\) is already a simplified number and does not have any exponents to write as positive or negative, we don't need to do anything else to express it with only positive exponents.
3. **Final Result**:
- Therefore, the entire expression simplifies to:
\[
1
\]
***Conclusion:***
The final answer is:
\[
1
\]
So, it seems there may have been a misunderstanding in the initial conditions, as this expression simplifies directly to the whole number \(1\). If you have any specific requirements for positive fractions, just let me know!
1. **Using the Zero Exponent Rule**:
- According to the property of exponents, any non-zero number raised to the power of 0 is equal to 1. Thus, we have:
\[
(14^3 \times 27^{-7})^0 = 1
\]
2. **Expressing with Positive Exponents**:
- Since \(1\) is already a simplified number and does not have any exponents to write as positive or negative, we don't need to do anything else to express it with only positive exponents.
3. **Final Result**:
- Therefore, the entire expression simplifies to:
\[
1
\]
***Conclusion:***
The final answer is:
\[
1
\]
So, it seems there may have been a misunderstanding in the initial conditions, as this expression simplifies directly to the whole number \(1\). If you have any specific requirements for positive fractions, just let me know!
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