Question

What is the property of integer exponents to generate equivalent expressions to (143^27−7)^34 with only positive exponents?

Answer 1

To rewrite the expression \((143^{27} - 7)^{34}\) using the property of integer exponents, we can apply the following rules of exponents:

1. **Distribution of Exponents**: When raising a power to a power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
2. **Positive Exponents**: If you wish to express the expression with positive exponents, ensure any negative exponents are re-expressed appropriately.

In our case, we do not have any negative exponents in the expression \((143^{27} - 7)^{34}\), so reworking them isn't necessary, as both \(143^{27}\) and \(7\) are positive.

Now, we can apply the power of a difference:

\[
(143^{27} - 7)^{34}
\]

This expression already contains only positive exponents for the base \(143\). There is no simplification that involves negative exponents that changes the meaning of this expression.

Thus, an equivalent expression using the properties of exponents is:

\[
(143^{27})^{34} - 7^{34}
\]

However, this is not correct because we cannot distribute exponents in a subtraction like we can with multiplication. Instead, the expression remains as initially presented.

Therefore, the optimal form remains:

\[
(143^{27} - 7)^{34}
\]

This keeps all exponents positive. Thus, fundamentally, the property of integer exponents simply affirms that in this expression, there are no changes required to achieve all positive exponents, and the expression itself is the final simplified version as it stands.