Asked by No name 🫦
Play the property of negative exponents to generate an expression equivalent to 3^-17/8^-14 with positive exponents only
3^-17/8^-14 = blank
3^-17/8^-14 = blank
Answers
Answered by
No name 🫦
R u sure
Answered by
No name 🫦
Apply the properties of integer exponents to generate equivalent expressions to (7^-3) ^7 * 7^20/ 6^-2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.
(7^-3) ^7 * 7^20/ 6^-2 = blank
(7^-3) ^7 * 7^20/ 6^-2 = blank
Answered by
No name 🫦
So its 36/7?
Answered by
No name 🫦
Simplify the expression(4x^3)^2 using the properties of integer exponents.
4x^6
16x^6
8x^6
16x^5
4x^6
16x^6
8x^6
16x^5
Answered by
No name 🫦
Play the properties of integer exponents to generate an equivalent, algebraic expression to (4x^4 y ^-4)^3/ (x^7y3)^2
Answered by
GPT-4o mini
To simplify the expression \( \frac{3^{-17}}{8^{-14}} \) using the property of negative exponents, we can rewrite the negative exponents as follows:
\[
\frac{3^{-17}}{8^{-14}} = 3^{-17} \cdot 8^{14}
\]
Next, we need to express \( 8^{14} \) using base 2, since \( 8 = 2^3 \):
\[
8^{14} = (2^3)^{14} = 2^{3 \cdot 14} = 2^{42}
\]
Now the original expression becomes:
\[
3^{-17} \cdot 8^{14} = 3^{-17} \cdot 2^{42}
\]
Next, we can express \( 3^{-17} \) as:
\[
3^{-17} = \frac{1}{3^{17}}
\]
So, the expression now looks like:
\[
\frac{2^{42}}{3^{17}}
\]
Therefore, the final expression with positive exponents only is:
\[
\frac{2^{42}}{3^{17}}
\]
Thus, your result is:
\[
\frac{2^{42}}{3^{17}}
\]
\[
\frac{3^{-17}}{8^{-14}} = 3^{-17} \cdot 8^{14}
\]
Next, we need to express \( 8^{14} \) using base 2, since \( 8 = 2^3 \):
\[
8^{14} = (2^3)^{14} = 2^{3 \cdot 14} = 2^{42}
\]
Now the original expression becomes:
\[
3^{-17} \cdot 8^{14} = 3^{-17} \cdot 2^{42}
\]
Next, we can express \( 3^{-17} \) as:
\[
3^{-17} = \frac{1}{3^{17}}
\]
So, the expression now looks like:
\[
\frac{2^{42}}{3^{17}}
\]
Therefore, the final expression with positive exponents only is:
\[
\frac{2^{42}}{3^{17}}
\]
Thus, your result is:
\[
\frac{2^{42}}{3^{17}}
\]
Answered by
GPT-4o mini
Let's carefully double-check the steps for converting \( \frac{3^{-17}}{8^{-14}} \) to an expression with only positive exponents.
1. Start with the expression:
\[
\frac{3^{-17}}{8^{-14}}
\]
2. Use the property of negative exponents:
\[
\frac{3^{-17}}{8^{-14}} = 3^{-17} \cdot 8^{14}
\]
3. Rewrite \( 8 \) as \( 2^3 \):
\[
8^{14} = (2^3)^{14} = 2^{3 \cdot 14} = 2^{42}
\]
4. Now substitute this back into the expression:
\[
3^{-17} \cdot 8^{14} = 3^{-17} \cdot 2^{42}
\]
5. Next, rewrite \( 3^{-17} \):
\[
3^{-17} = \frac{1}{3^{17}}
\]
6. Therefore, combining everything gives:
\[
\frac{2^{42}}{3^{17}}
\]
So, the correct final expression with only positive exponents is indeed:
\[
\frac{2^{42}}{3^{17}}
\]
Yes, I am confident in this result. If you have any further questions or if there's anything specific you would like to clarify, please let me know!
1. Start with the expression:
\[
\frac{3^{-17}}{8^{-14}}
\]
2. Use the property of negative exponents:
\[
\frac{3^{-17}}{8^{-14}} = 3^{-17} \cdot 8^{14}
\]
3. Rewrite \( 8 \) as \( 2^3 \):
\[
8^{14} = (2^3)^{14} = 2^{3 \cdot 14} = 2^{42}
\]
4. Now substitute this back into the expression:
\[
3^{-17} \cdot 8^{14} = 3^{-17} \cdot 2^{42}
\]
5. Next, rewrite \( 3^{-17} \):
\[
3^{-17} = \frac{1}{3^{17}}
\]
6. Therefore, combining everything gives:
\[
\frac{2^{42}}{3^{17}}
\]
So, the correct final expression with only positive exponents is indeed:
\[
\frac{2^{42}}{3^{17}}
\]
Yes, I am confident in this result. If you have any further questions or if there's anything specific you would like to clarify, please let me know!
Answered by
GPT-4o mini
To simplify the expression \( \frac{(7^{-3})^7 \cdot 7^{20}}{6^{-2}} \) and convert it into an equivalent expression with only positive exponents, let's follow these steps:
1. **First, simplify \( (7^{-3})^7 \)**:
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \):
\[
(7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21}
\]
2. **Now multiply by \( 7^{20} \)**:
\[
7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1}
\]
3. **Now we have the current expression**:
\[
\frac{7^{-1}}{6^{-2}}
\]
4. **Convert the negative exponent in the denominator**:
\[
\frac{7^{-1}}{6^{-2}} = 7^{-1} \cdot 6^2
\]
5. **Now apply the property of negative exponents again**:
\[
7^{-1} = \frac{1}{7}
\]
Therefore:
\[
7^{-1} \cdot 6^2 = \frac{6^2}{7} = \frac{36}{7}
\]
So, the simplified expression with only positive exponents is:
\[
\frac{36}{7}
\]
### Final Answer
The equivalent expression is:
\[
\frac{36}{7}
\]
This is in the form of an improper fraction as requested. If you have any further questions or need any additional assistance, feel free to ask!
1. **First, simplify \( (7^{-3})^7 \)**:
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \):
\[
(7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21}
\]
2. **Now multiply by \( 7^{20} \)**:
\[
7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1}
\]
3. **Now we have the current expression**:
\[
\frac{7^{-1}}{6^{-2}}
\]
4. **Convert the negative exponent in the denominator**:
\[
\frac{7^{-1}}{6^{-2}} = 7^{-1} \cdot 6^2
\]
5. **Now apply the property of negative exponents again**:
\[
7^{-1} = \frac{1}{7}
\]
Therefore:
\[
7^{-1} \cdot 6^2 = \frac{6^2}{7} = \frac{36}{7}
\]
So, the simplified expression with only positive exponents is:
\[
\frac{36}{7}
\]
### Final Answer
The equivalent expression is:
\[
\frac{36}{7}
\]
This is in the form of an improper fraction as requested. If you have any further questions or need any additional assistance, feel free to ask!
Answered by
GPT-4o mini
Yes, that's correct! The expression
\[
\frac{(7^{-3})^7 \cdot 7^{20}}{6^{-2}}
\]
simplifies to
\[
\frac{36}{7}
\]
This is an improper fraction, and it is indeed the final answer. If you have any more questions or need further clarification, feel free to ask!
\[
\frac{(7^{-3})^7 \cdot 7^{20}}{6^{-2}}
\]
simplifies to
\[
\frac{36}{7}
\]
This is an improper fraction, and it is indeed the final answer. If you have any more questions or need further clarification, feel free to ask!
Answered by
GPT-4o mini
To simplify the expression \((4x^3)^2\) using the properties of integer exponents, we can apply the following rules:
1. **Power of a product**: \((ab)^n = a^n b^n\)
2. **Power of a power**: \((a^m)^n = a^{m \cdot n}\)
Now, applying these rules to the expression:
\[
(4x^3)^2 = 4^2 \cdot (x^3)^2
\]
Calculating each part:
- \(4^2 = 16\)
- \((x^3)^2 = x^{3 \cdot 2} = x^6\)
Now we combine these results:
\[
(4x^3)^2 = 16x^6
\]
So, the simplified expression is:
\[
\boxed{16x^6}
\]
Thus, the correct answer is **16x^6**.
1. **Power of a product**: \((ab)^n = a^n b^n\)
2. **Power of a power**: \((a^m)^n = a^{m \cdot n}\)
Now, applying these rules to the expression:
\[
(4x^3)^2 = 4^2 \cdot (x^3)^2
\]
Calculating each part:
- \(4^2 = 16\)
- \((x^3)^2 = x^{3 \cdot 2} = x^6\)
Now we combine these results:
\[
(4x^3)^2 = 16x^6
\]
So, the simplified expression is:
\[
\boxed{16x^6}
\]
Thus, the correct answer is **16x^6**.
Answered by
GPT-4o mini
To simplify the expression
\[
\frac{(4x^4 y^{-4})^3}{(x^7 y^3)^2},
\]
we will apply the properties of integer exponents step-by-step.
### Step 1: Simplify the numerator
Start with the numerator:
\[
(4x^4 y^{-4})^3.
\]
Using the property of exponents \((a^m)^n = a^{mn}\) and the power of a product property, we get:
\[
= 4^3 (x^4)^3 (y^{-4})^3 = 64 x^{12} y^{-12}.
\]
### Step 2: Simplify the denominator
Now, simplify the denominator:
\[
(x^7 y^3)^2.
\]
Using the same exponent properties, we have:
\[
= (x^7)^2 (y^3)^2 = x^{14} y^6.
\]
### Step 3: Combine the fractions
Now we can substitute these results back into the expression:
\[
\frac{64 x^{12} y^{-12}}{x^{14} y^6}.
\]
### Step 4: Simplify the fraction
Now we will simplify this fraction using the properties of exponents:
\[
= 64 \cdot \frac{x^{12}}{x^{14}} \cdot \frac{y^{-12}}{y^6}.
\]
For the \(x\) terms, use \( \frac{a^m}{a^n} = a^{m-n} \):
\[
= 64 \cdot x^{12-14} \cdot y^{-12-6} = 64 \cdot x^{-2} \cdot y^{-18}.
\]
### Step 5: Rewrite with positive exponents
To express this with only positive exponents, recall that \(a^{-n} = \frac{1}{a^n}\):
\[
= \frac{64}{x^2 y^{18}}.
\]
Thus, the final simplified equivalent algebraic expression is:
\[
\boxed{\frac{64}{x^2 y^{18}}}.
\]
\[
\frac{(4x^4 y^{-4})^3}{(x^7 y^3)^2},
\]
we will apply the properties of integer exponents step-by-step.
### Step 1: Simplify the numerator
Start with the numerator:
\[
(4x^4 y^{-4})^3.
\]
Using the property of exponents \((a^m)^n = a^{mn}\) and the power of a product property, we get:
\[
= 4^3 (x^4)^3 (y^{-4})^3 = 64 x^{12} y^{-12}.
\]
### Step 2: Simplify the denominator
Now, simplify the denominator:
\[
(x^7 y^3)^2.
\]
Using the same exponent properties, we have:
\[
= (x^7)^2 (y^3)^2 = x^{14} y^6.
\]
### Step 3: Combine the fractions
Now we can substitute these results back into the expression:
\[
\frac{64 x^{12} y^{-12}}{x^{14} y^6}.
\]
### Step 4: Simplify the fraction
Now we will simplify this fraction using the properties of exponents:
\[
= 64 \cdot \frac{x^{12}}{x^{14}} \cdot \frac{y^{-12}}{y^6}.
\]
For the \(x\) terms, use \( \frac{a^m}{a^n} = a^{m-n} \):
\[
= 64 \cdot x^{12-14} \cdot y^{-12-6} = 64 \cdot x^{-2} \cdot y^{-18}.
\]
### Step 5: Rewrite with positive exponents
To express this with only positive exponents, recall that \(a^{-n} = \frac{1}{a^n}\):
\[
= \frac{64}{x^2 y^{18}}.
\]
Thus, the final simplified equivalent algebraic expression is:
\[
\boxed{\frac{64}{x^2 y^{18}}}.
\]
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