To simplify the given expression, we need to perform a few straightforward steps:
[tex]\[ \frac{x \cdot x^2 \cdot x^3 \cdot x^4 \cdot x^5}{x^8 \cdot x^{15}} \][/tex]
First, let's simplify the numerator and the denominator by combining the exponents of [tex]\( x \)[/tex].
Step 1: Simplify the numerator:
The numerator is [tex]\( x \cdot x^2 \cdot x^3 \cdot x^4 \cdot x^5 \)[/tex].
By using the properties of exponents (specifically, [tex]\( x^a \cdot x^b = x^{a+b} \)[/tex]), we can combine these into a single term:
[tex]\[ (x \cdot x^2 \cdot x^3 \cdot x^4 \cdot x^5) = x^{1+2+3+4+5} \][/tex]
Summing the exponents, we get:
[tex]\[ 1 + 2 + 3 + 4 + 5 = 15 \][/tex]
Thus, the numerator simplifies to:
[tex]\[ x^{15} \][/tex]
Step 2: Simplify the denominator:
The denominator is [tex]\( x^8 \cdot x^{15} \)[/tex].
Similarly, we combine these exponents:
[tex]\[ (x^8 \cdot x^{15}) = x^{8+15} \][/tex]
Summing the exponents, we get:
[tex]\[ 8 + 15 = 23 \][/tex]
Thus, the denominator simplifies to:
[tex]\[ x^{23} \][/tex]
Step 3: Simplify the expression:
Now, we have the simplified expression:
[tex]\[ \frac{x^{15}}{x^{23}} \][/tex]
Using the properties of exponents, specifically [tex]\( \frac{x^a}{x^b} = x^{a-b} \)[/tex], we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[ x^{15-23} = x^{-8} \][/tex]
Thus, the final simplified form of the expression is:
[tex]\[ x^{-8} \][/tex]
Therefore, the correct answer is:
B) [tex]\( x^{-8} \)[/tex]
