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Original Question
Expand the properties of exponents to involve monomial bases with integer exponents to simplify the expression (-25x^(8))/(5x)....Asked by Milana
Expand the properties of exponents to involve monomial bases with integer exponents to simplify the expression -25x^8/5x
Answers
Answered by
Milana
Use the properties of integer exponents to simplify the expression (-4a^4) (-2a^2)
*8a^2
8a^6
-8a^6
8a^8
*8a^2
8a^6
-8a^6
8a^8
Answered by
Milana
Apply the properties of integer exponents to generate an equivalent algebraic expression to 3x^3y^-4/12x^7y^4
Answered by
Milana
Apply the properties of integer exponents to simplify the algebraic expression (2xy^3)^4 * (x^2y)^3*
Answered by
Milana
Apply the properties of integer exponents to simplify the algebraic expression (-5a^3b^4c^-2/3a^-4b^0c^3) 2
Answered by
Milana
Milana
answered
7 seconds ago
Apply the properties of integer exponents to simplify the algebraic expression (-5a^3b^4c^-2/3a^-4b^0c^3) 2
C^10/a^14b^8
9/25a^12
-5c^10/3a^14b^8
9c^10/25a^14b^8
answered
7 seconds ago
Apply the properties of integer exponents to simplify the algebraic expression (-5a^3b^4c^-2/3a^-4b^0c^3) 2
C^10/a^14b^8
9/25a^12
-5c^10/3a^14b^8
9c^10/25a^14b^8
Answered by
GPT-4o mini
To simplify the expression \(-25x^8 / 5x\) using the properties of exponents, we can follow these steps:
1. **Divide the coefficients**: The coefficients are \(-25\) and \(5\). Dividing these gives:
\[
-25 \div 5 = -5
\]
2. **Apply the properties of exponents to the variable \(x\)**:
We can use the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \). In this case, we have \(x^8\) in the numerator and \(x^1\) (which is just \(x\)) in the denominator:
\[
\frac{x^8}{x^1} = x^{8-1} = x^7
\]
3. **Combine the results**:
Now that we have simplified both the coefficients and the variables, we can combine them:
\[
-25x^8 / 5x = -5x^7
\]
Thus, the simplified expression is:
\[
\boxed{-5x^7}
\]
1. **Divide the coefficients**: The coefficients are \(-25\) and \(5\). Dividing these gives:
\[
-25 \div 5 = -5
\]
2. **Apply the properties of exponents to the variable \(x\)**:
We can use the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \). In this case, we have \(x^8\) in the numerator and \(x^1\) (which is just \(x\)) in the denominator:
\[
\frac{x^8}{x^1} = x^{8-1} = x^7
\]
3. **Combine the results**:
Now that we have simplified both the coefficients and the variables, we can combine them:
\[
-25x^8 / 5x = -5x^7
\]
Thus, the simplified expression is:
\[
\boxed{-5x^7}
\]
Answered by
GPT-4o mini
To simplify the expression \((-4a^4)(-2a^2)\), we can follow these steps:
1. **Multiply the coefficients**:
\[
(-4) \times (-2) = 8
\]
2. **Apply the properties of exponents to the variable \(a\)**:
We use the property that states \(a^m \times a^n = a^{m+n}\). In this case, we have:
\[
a^4 \times a^2 = a^{4+2} = a^6
\]
3. **Combine the results**:
Now we can combine the multiplied coefficients and variables:
\[
(-4a^4)(-2a^2) = 8a^6
\]
Thus, the simplified expression is:
\[
\boxed{8a^6}
\]
1. **Multiply the coefficients**:
\[
(-4) \times (-2) = 8
\]
2. **Apply the properties of exponents to the variable \(a\)**:
We use the property that states \(a^m \times a^n = a^{m+n}\). In this case, we have:
\[
a^4 \times a^2 = a^{4+2} = a^6
\]
3. **Combine the results**:
Now we can combine the multiplied coefficients and variables:
\[
(-4a^4)(-2a^2) = 8a^6
\]
Thus, the simplified expression is:
\[
\boxed{8a^6}
\]
Answered by
GPT-4o mini
To simplify the expression \(\frac{3x^3y^{-4}}{12x^7y^4}\) using the properties of integer exponents, we can follow these steps:
1. **Divide the coefficients**:
\[
\frac{3}{12} = \frac{1}{4}
\]
2. **Apply the properties of exponents to \(x\)**:
Using the property \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{x^3}{x^7} = x^{3-7} = x^{-4}
\]
3. **Apply the properties of exponents to \(y\)**:
Using the same property:
\[
\frac{y^{-4}}{y^4} = y^{-4-4} = y^{-8}
\]
4. **Combine the results**:
Now we can combine everything:
\[
\frac{3x^3y^{-4}}{12x^7y^4} = \frac{1}{4}x^{-4}y^{-8}
\]
The expression can also be rewritten using positive exponents:
\[
\frac{1}{4} \cdot \frac{1}{x^4} \cdot \frac{1}{y^8} = \frac{1}{4x^4y^8}
\]
Thus, the equivalent algebraic expression is:
\[
\boxed{\frac{1}{4x^4y^8}}
\]
1. **Divide the coefficients**:
\[
\frac{3}{12} = \frac{1}{4}
\]
2. **Apply the properties of exponents to \(x\)**:
Using the property \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{x^3}{x^7} = x^{3-7} = x^{-4}
\]
3. **Apply the properties of exponents to \(y\)**:
Using the same property:
\[
\frac{y^{-4}}{y^4} = y^{-4-4} = y^{-8}
\]
4. **Combine the results**:
Now we can combine everything:
\[
\frac{3x^3y^{-4}}{12x^7y^4} = \frac{1}{4}x^{-4}y^{-8}
\]
The expression can also be rewritten using positive exponents:
\[
\frac{1}{4} \cdot \frac{1}{x^4} \cdot \frac{1}{y^8} = \frac{1}{4x^4y^8}
\]
Thus, the equivalent algebraic expression is:
\[
\boxed{\frac{1}{4x^4y^8}}
\]
Answered by
GPT-4o mini
To simplify the expression \((2xy^3)^4 \cdot (x^2y)^3\) using the properties of integer exponents, we can follow these steps:
### Step 1: Apply the Power of a Product Property
1. **Simplify \((2xy^3)^4\)**:
Use the property \((ab)^n = a^n b^n\):
\[
(2xy^3)^4 = 2^4 \cdot x^4 \cdot (y^3)^4 = 16x^4y^{12}
\]
2. **Simplify \((x^2y)^3\)**:
Similarly:
\[
(x^2y)^3 = (x^2)^3 \cdot y^3 = x^{2 \cdot 3} \cdot y^3 = x^6y^3
\]
### Step 2: Combine the Results
Now we can multiply the simplified expressions:
\[
(16x^4y^{12}) \cdot (x^6y^3)
\]
### Step 3: Apply the Product of Powers Property
1. **Multiply the coefficients**:
\[
16 \cdot 1 = 16
\]
2. **Combine the \(x\) terms**:
\[
x^4 \cdot x^6 = x^{4+6} = x^{10}
\]
3. **Combine the \(y\) terms**:
\[
y^{12} \cdot y^3 = y^{12+3} = y^{15}
\]
### Final Result
Putting it all together, we get:
\[
16x^{10}y^{15}
\]
Thus, the simplified expression is:
\[
\boxed{16x^{10}y^{15}}
\]
### Step 1: Apply the Power of a Product Property
1. **Simplify \((2xy^3)^4\)**:
Use the property \((ab)^n = a^n b^n\):
\[
(2xy^3)^4 = 2^4 \cdot x^4 \cdot (y^3)^4 = 16x^4y^{12}
\]
2. **Simplify \((x^2y)^3\)**:
Similarly:
\[
(x^2y)^3 = (x^2)^3 \cdot y^3 = x^{2 \cdot 3} \cdot y^3 = x^6y^3
\]
### Step 2: Combine the Results
Now we can multiply the simplified expressions:
\[
(16x^4y^{12}) \cdot (x^6y^3)
\]
### Step 3: Apply the Product of Powers Property
1. **Multiply the coefficients**:
\[
16 \cdot 1 = 16
\]
2. **Combine the \(x\) terms**:
\[
x^4 \cdot x^6 = x^{4+6} = x^{10}
\]
3. **Combine the \(y\) terms**:
\[
y^{12} \cdot y^3 = y^{12+3} = y^{15}
\]
### Final Result
Putting it all together, we get:
\[
16x^{10}y^{15}
\]
Thus, the simplified expression is:
\[
\boxed{16x^{10}y^{15}}
\]
Answered by
GPT-4o mini
To simplify the expression \(\left(-\frac{5a^3b^4c^{-2}}{3a^{-4}b^0c^3}\right) 2\), we can follow these steps:
### Step 1: Simplify the Fraction
1. **Separate the coefficients**:
\[
-\frac{5}{3}
\]
2. **Apply the properties of exponents to the variable \(a\)**:
Using the property \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{a^3}{a^{-4}} = a^{3 - (-4)} = a^{3 + 4} = a^7
\]
3. **Apply the properties of exponents to the variable \(b\)**:
Since \(b^0 = 1\):
\[
\frac{b^4}{b^0} = \frac{b^4}{1} = b^4
\]
4. **Apply the properties of exponents to the variable \(c\)**:
\[
\frac{c^{-2}}{c^3} = c^{-2 - 3} = c^{-5}
\]
### Step 2: Combine Everything
So combining the results from above:
\[
-\frac{5}{3} a^7 b^4 c^{-5}
\]
### Step 3: Incorporate the Multiplication by 2
Now, we multiply the entire expression by 2:
\[
2 \left(-\frac{5}{3} a^7 b^4 c^{-5}\right) = -\frac{10}{3} a^7 b^4 c^{-5}
\]
### Step 4: Rewrite Using Positive Exponents
To express \(c^{-5}\) with a positive exponent, we write it as:
\[
\frac{-10}{3} a^7 b^4 \frac{1}{c^5} = -\frac{10}{3} \frac{a^7 b^4}{c^5}
\]
### Final Result
Thus, the simplified expression is:
\[
\boxed{-\frac{10}{3} \frac{a^7 b^4}{c^5}}
\]
### Step 1: Simplify the Fraction
1. **Separate the coefficients**:
\[
-\frac{5}{3}
\]
2. **Apply the properties of exponents to the variable \(a\)**:
Using the property \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{a^3}{a^{-4}} = a^{3 - (-4)} = a^{3 + 4} = a^7
\]
3. **Apply the properties of exponents to the variable \(b\)**:
Since \(b^0 = 1\):
\[
\frac{b^4}{b^0} = \frac{b^4}{1} = b^4
\]
4. **Apply the properties of exponents to the variable \(c\)**:
\[
\frac{c^{-2}}{c^3} = c^{-2 - 3} = c^{-5}
\]
### Step 2: Combine Everything
So combining the results from above:
\[
-\frac{5}{3} a^7 b^4 c^{-5}
\]
### Step 3: Incorporate the Multiplication by 2
Now, we multiply the entire expression by 2:
\[
2 \left(-\frac{5}{3} a^7 b^4 c^{-5}\right) = -\frac{10}{3} a^7 b^4 c^{-5}
\]
### Step 4: Rewrite Using Positive Exponents
To express \(c^{-5}\) with a positive exponent, we write it as:
\[
\frac{-10}{3} a^7 b^4 \frac{1}{c^5} = -\frac{10}{3} \frac{a^7 b^4}{c^5}
\]
### Final Result
Thus, the simplified expression is:
\[
\boxed{-\frac{10}{3} \frac{a^7 b^4}{c^5}}
\]
Answered by
GPT-4o mini
To simplify the expression \(\left(-\frac{5a^3b^4c^{-2}}{3a^{-4}b^0c^3}\right) \cdot 2\), we'll follow the steps.
### Step 1: Simplify the Fraction
1. **Separate the coefficients**:
\[
-\frac{5}{3}
\]
2. **Apply the properties of exponents to the variable \(a\)**:
Using \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{a^3}{a^{-4}} = a^{3 - (-4)} = a^{3 + 4} = a^7
\]
3. **Apply the properties of exponents to the variable \(b\)**:
Since \(b^0 = 1\):
\[
\frac{b^4}{b^0} = \frac{b^4}{1} = b^4
\]
4. **Apply the properties of exponents to the variable \(c\)**:
\[
\frac{c^{-2}}{c^3} = c^{-2 - 3} = c^{-5}
\]
Combining these results, we get:
\[
-\frac{5}{3} a^7 b^4 c^{-5}
\]
### Step 2: Incorporate the Multiplication by 2
We now multiply the entire expression by 2:
\[
2 \left(-\frac{5}{3} a^7 b^4 c^{-5}\right) = -\frac{10}{3} a^7 b^4 c^{-5}
\]
### Step 3: Rewrite Using Positive Exponents
To express \(c^{-5}\) with a positive exponent:
\[
-\frac{10}{3} a^7 b^4 \cdot \frac{1}{c^5} = -\frac{10 a^7 b^4}{3c^5}
\]
### Check for Matching Options
Now, let's see how the simplified result can match the provided options:
1. **\(c^{10}/a^{14}b^{8}\)**
2. **\(9/25a^{12}\)**
3. **\(-5c^{10}/3a^{14}b^{8}\)**
4. **\(9c^{10}/25a^{14}b^{8}\)**
Our final expression is:
\[
-\frac{10 a^7 b^4}{3c^5}
\]
### Step 4: Analyze Given Choices
The options seem to not match directly, as none of the options resemble \(-\frac{10 a^7 b^4}{3 c^5}\). However, let’s compare our characteristics with the given options:
- The exponentials of \(a\) and \(b\) values, as well as those of \(c\), might suggest a relation, but the coefficients and exponent forms differ.
- Check if \(c^{-5}\) could potentially relate to \(c^{10}\) in terms of multiplying or similar structure through negatives; this does not hold.
### Conclusion
It appears that the expression we derived doesn't match any of the provided choices. The simplified expression is \(-\frac{10 a^7 b^4}{3 c^5}\). If a typo occurred in the answer options or additional constraints not provided initially were included, we cannot draw a correspondence with those choices beneficially.
Thus, the accurate simplified algebraic expression remains:
\[
\boxed{-\frac{10 a^7 b^4}{3 c^5}}
\]
### Step 1: Simplify the Fraction
1. **Separate the coefficients**:
\[
-\frac{5}{3}
\]
2. **Apply the properties of exponents to the variable \(a\)**:
Using \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{a^3}{a^{-4}} = a^{3 - (-4)} = a^{3 + 4} = a^7
\]
3. **Apply the properties of exponents to the variable \(b\)**:
Since \(b^0 = 1\):
\[
\frac{b^4}{b^0} = \frac{b^4}{1} = b^4
\]
4. **Apply the properties of exponents to the variable \(c\)**:
\[
\frac{c^{-2}}{c^3} = c^{-2 - 3} = c^{-5}
\]
Combining these results, we get:
\[
-\frac{5}{3} a^7 b^4 c^{-5}
\]
### Step 2: Incorporate the Multiplication by 2
We now multiply the entire expression by 2:
\[
2 \left(-\frac{5}{3} a^7 b^4 c^{-5}\right) = -\frac{10}{3} a^7 b^4 c^{-5}
\]
### Step 3: Rewrite Using Positive Exponents
To express \(c^{-5}\) with a positive exponent:
\[
-\frac{10}{3} a^7 b^4 \cdot \frac{1}{c^5} = -\frac{10 a^7 b^4}{3c^5}
\]
### Check for Matching Options
Now, let's see how the simplified result can match the provided options:
1. **\(c^{10}/a^{14}b^{8}\)**
2. **\(9/25a^{12}\)**
3. **\(-5c^{10}/3a^{14}b^{8}\)**
4. **\(9c^{10}/25a^{14}b^{8}\)**
Our final expression is:
\[
-\frac{10 a^7 b^4}{3c^5}
\]
### Step 4: Analyze Given Choices
The options seem to not match directly, as none of the options resemble \(-\frac{10 a^7 b^4}{3 c^5}\). However, let’s compare our characteristics with the given options:
- The exponentials of \(a\) and \(b\) values, as well as those of \(c\), might suggest a relation, but the coefficients and exponent forms differ.
- Check if \(c^{-5}\) could potentially relate to \(c^{10}\) in terms of multiplying or similar structure through negatives; this does not hold.
### Conclusion
It appears that the expression we derived doesn't match any of the provided choices. The simplified expression is \(-\frac{10 a^7 b^4}{3 c^5}\). If a typo occurred in the answer options or additional constraints not provided initially were included, we cannot draw a correspondence with those choices beneficially.
Thus, the accurate simplified algebraic expression remains:
\[
\boxed{-\frac{10 a^7 b^4}{3 c^5}}
\]