Question
Apply the properties of integer exponents to simplify the algebraic expression and it is in parentheses 2XY small three on top of the Y and then we have outside of the parentheses a small for parentheses and with a small on top and why and then outside of the parentheses we had a three on top With the answer be on top small 15 on top or would it be 16 X with a small 25 on top or 32 X with a small tan on top and white with a small 11 on top or 16 X with a small tan on top with a small 15 on top
Answers
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Answered by
GPT-4o mini
To simplify the expression you provided, we need to carefully interpret the notation. From your description, it seems like we have the expression:
\[
(2XY^3)^4
\]
This expression is raised to the power of 4, and we can apply the properties of exponents to simplify it.
Using the property \((ab)^n = a^n b^n\) and \((x^m)^n = x^{m \cdot n}\), we can break down the expression as follows:
1. Raise the coefficient and each variable inside the parentheses to the power of 4:
\[
(2XY^3)^4 = 2^4 \cdot X^4 \cdot (Y^3)^4
\]
2. Calculate \(2^4\):
\[
2^4 = 16
\]
3. For \(X^4\), it remains \(X^4\).
4. For \(Y^3\) raised to the power of 4, we use the property \((Y^3)^4 = Y^{3 \cdot 4} = Y^{12}\).
Putting it all together, we have:
\[
(2XY^3)^4 = 16X^4Y^{12}
\]
If we wanted to express this with some additional notation you mentioned, but based on the final result, the expression simplifies to:
\[
16X^4Y^{12}
\]
As for the suggested options you mentioned like \(16X^{25}\), \(32X^{10}Y^{11}\), or \(16X^{10}\) with the numerator seemingly being some form, they don't align with the simplified expression we've derived from the exponentiation.
Thus, the correct answer is:
\[
\boxed{16X^4Y^{12}}
\]
\[
(2XY^3)^4
\]
This expression is raised to the power of 4, and we can apply the properties of exponents to simplify it.
Using the property \((ab)^n = a^n b^n\) and \((x^m)^n = x^{m \cdot n}\), we can break down the expression as follows:
1. Raise the coefficient and each variable inside the parentheses to the power of 4:
\[
(2XY^3)^4 = 2^4 \cdot X^4 \cdot (Y^3)^4
\]
2. Calculate \(2^4\):
\[
2^4 = 16
\]
3. For \(X^4\), it remains \(X^4\).
4. For \(Y^3\) raised to the power of 4, we use the property \((Y^3)^4 = Y^{3 \cdot 4} = Y^{12}\).
Putting it all together, we have:
\[
(2XY^3)^4 = 16X^4Y^{12}
\]
If we wanted to express this with some additional notation you mentioned, but based on the final result, the expression simplifies to:
\[
16X^4Y^{12}
\]
As for the suggested options you mentioned like \(16X^{25}\), \(32X^{10}Y^{11}\), or \(16X^{10}\) with the numerator seemingly being some form, they don't align with the simplified expression we've derived from the exponentiation.
Thus, the correct answer is:
\[
\boxed{16X^4Y^{12}}
\]
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