Sure, let's solve the given expression step-by-step to understand the simplification process.
We start with the expression:
[tex]\[
\frac{X^{p-q+1} \times X^{q-r+1} \times X^{ p -p+1}}{X^3}
\][/tex]
### Step 1: Combine the Exponents in the Numerator
First, we need to simplify the expression in the numerator. Since we have a product of terms with the same base [tex]\(X\)[/tex], we can add their exponents:
[tex]\[
X^{p-q+1} \times X^{q-r+1} \times X^{p-p+1} = X^{(p-q+1) + (q-r+1) + (p-p+1)}
\][/tex]
### Step 2: Simplify the Exponents in the Numerator
Let's combine and simplify the exponents:
[tex]\[
(p - q + 1) + (q - r + 1) + (p - p + 1)
\][/tex]
Combine like terms:
[tex]\[
= p - q + q - r + p - p + 1 + 1 + 1
\][/tex]
Notice that [tex]\( -q \)[/tex] and [tex]\( +q \)[/tex] cancel out, and [tex]\( +p \)[/tex] and [tex]\( -p \)[/tex] cancel out:
[tex]\[
= p - r + 3
\][/tex]
So the expression simplifies to:
[tex]\[
\frac{X^{p-r+3}}{X^3}
\][/tex]
### Step 3: Simplify the Entire Expression
We now have a single term in the numerator and a single term in the denominator, both with base [tex]\(X\)[/tex]. We can subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
= X^{(p-r+3) - 3}
\][/tex]
Simplify the exponent:
[tex]\[
= X^{p-r}
\][/tex]
### Final Answer
So the given expression simplifies to:
[tex]\[
X^{p - r}
\][/tex]
This is the simplified form of the given expression.
