Question

Simplify using the laws of exponents. -1 25 O-5 O1/125 O-1/5 O-125

Answer 1

To simplify the expression \(-1 \cdot 25 \cdot O^{-5} \cdot O^{1/125} \cdot O^{-1/5} \cdot O^{-125}\), we can use the laws of exponents, specifically the product of powers rule, which states that \(a^m \cdot a^n = a^{m+n}\).

Let's combine the exponents of \(O\):

1. Identify the exponents:
- The exponents of \(O\) are \(-5\), \(\frac{1}{125}\), \(-\frac{1}{5}\), and \(-125\).

2. Add the exponents:
\[
-5 + \frac{1}{125} - \frac{1}{5} - 125
\]

To add these, we need a common denominator. The common denominator for \(125\) and \(5\) is \(125\). Therefore, we can rewrite \(-\frac{1}{5}\) as:

\[
-\frac{1}{5} = -\frac{25}{125}
\]

Now we can add all the exponents together:

\[
-5 - 125 = -130
\]

Now converting \(-130\) to be over \(125\):
\[
-130 = -\frac{130 \cdot 125}{125} = -\frac{16250}{125}
\]
So we can write:
\[
-130 + \frac{1}{125} - \frac{25}{125} = -\frac{16250}{125} + \frac{1}{125} - \frac{25}{125}
\]
\[
= -\frac{16250 + 25 - 1}{125} = -\frac{16250 + 24}{125} = -\frac{16274}{125}
\]

Thus combining, the exponent sum is:
\[
-\frac{16274}{125}
\]

Therefore the expression simplifies to:
\[
-1 \cdot 25 \cdot O^{-\frac{16274}{125}} = -25 \cdot O^{-\frac{16274}{125}}
\]

And as a final result, we can write it as:
\[
-\frac{25}{O^{\frac{16274}{125}}}
\]

Thus, the simplified expression is:
\[
-\frac{25}{O^{\frac{16274}{125}}}
\]