To simplify the given polynomial expression, we will go through each term step by step:
-
First term: \((6x^3y^9)(7xy^2)\)
- Multiply the coefficients: \(6 \times 7 = 42\)
- Add the exponents of \(x\): \(3 + 1 = 4\)
- Add the exponents of \(y\): \(9 + 2 = 11\)
- This term simplifies to \(42x^4y^{11}\).
-
Second term: \((9x^3y^2)(3xy^9)\)
- Multiply the coefficients: \(9 \times 3 = 27\)
- Add the exponents of \(x\): \(3 + 1 = 4\)
- Add the exponents of \(y\): \(2 + 9 = 11\)
- This term simplifies to \(27x^4y^{11}\).
- Since this term is subtracted, it will become \(-27x^4y^{11}\).
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Third term: \((8xy^2)^2\)
- First, calculate \(8^2 = 64\).
- Multiply the coefficients: \(64\).
- Add the exponents of \(x\): \(1 \cdot 2 = 2\)
- Add the exponents of \(y\): \(2 \cdot 2 = 4\)
- This term simplifies to \(64x^2y^{4}\).
Now, we combine all three simplified terms together: \[ 42x^4y^{11} - 27x^4y^{11} + 64x^2y^{4} \] Combine the like terms for \(x^4y^{11}\): \[ (42 - 27)x^4y^{11} + 64x^2y^{4} = 15x^4y^{11} + 64x^2y^{4} \]
Thus, the overall simplified expression is: \[ 15x^4y^{11} + 64x^2y^{4} \]
Next, we compare the equivalent expressions provided:
- \(15x^4y^{11} + 8x^2y^{4}\) - Not equivalent
- \(-15x^4y^{11} + 64x^2y^{4}\) - Not equivalent
- \(42x^4y^{11} - 27x^4y^{11} + 64x^2y^{4}\) - Equivalent, as it simplifies back to the same expression.
- \(15x^4y^{11} + 64x^2y^{4}\) - Equivalent
- \(42x^3y^{9} - 27x^4y^{2} + 64x^{22}y^{4}\) - Not equivalent
Hence, the equivalent expressions are: 3 and 4.
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