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Expand the properties of exponents to involve monomial bases with integer exponents to simplify the expression -25x^8/5x

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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: \[ \boxed{-5x^7} \]

Use the properties of integer exponents to simplify the expression (-4a^4) (-2a^2)

*8a^2

8a^6

-8a^6

8a^8

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: \[ \boxed{8a^6} \]

Apply the properties of integer exponents to generate an equivalent algebraic expression to 3x^3y^-4/12x^7y^4

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: \[ \boxed{\frac{1}{4x^4y^8}} \]

Apply the properties of integer exponents to simplify the algebraic expression (2xy^3)^4 * (x^2y)^3*

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: \[ \boxed{16x^{10}y^{15}} \]

Apply the properties of integer exponents to simplify the algebraic expression (-5a^3b^4c^-2/3a^-4b^0c^3) 2

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: \[ \boxed{-\frac{10}{3} \frac{a^7 b^4}{c^5}} \]

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

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: \[ \boxed{-\frac{10 a^7 b^4}{3 c^5}} \]