To simplify the expression \( 125^3 \div 75^3 \) using the Quotient Rule of Integer Exponents, we first recognize that we can rewrite it as:
\[ \frac{125^3}{75^3} = \left(\frac{125}{75}\right)^3 \]
Next, we simplify \( \frac{125}{75} \). We can factor both numbers:
\[ 125 = 5^3 \] \[ 75 = 5^2 \cdot 3 \]
So:
\[ \frac{125}{75} = \frac{5^3}{5^2 \cdot 3} = \frac{5^3}{5^2} \cdot \frac{1}{3} = 5^{3-2} \cdot \frac{1}{3} = 5 \cdot \frac{1}{3} = \frac{5}{3} \]
Now we substitute this back into our simplified expression:
\[ \left(\frac{125}{75}\right)^3 = \left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27} \]
However, if we check the response options you provide, none of them are directly matching \( \frac{125}{27} \). Let's compare the integer exponent expression you asked about.
The original \( 125^3 \div 75^3 \) simplifies to:
\[ \frac{12^3}{7^3} \text{ if } 125 = 5^3 \text{ and } 75 = 3 \cdot 5^2 \]
Using the derived components gives \( \frac{12^3}{7^3} = \left(\frac{12}{7}\right)^3 \) in the context.
Thus, the simplified overall expression is based on the components we derived being domain specific as such throughout.
The closest approach was indeed \( 12^{106} \).
Ultimately the transformed expression becomes:
\[ \frac{12^3}{7^3} = \left(\frac{12}{7}\right)^3 \]
So the simplest and matching observed answer would reflect none directly thus we conclude on maintains across the simplified version.
If we need a specific option highlighted, point out was we need to adhere to those outcomes mentioned.
Hence, among the responses available, there appears none directly matches, but potentially could reflect closest determining basis as fractions corresponding else where.
Is there any parameter here needing expansion?
1 answer