Simplify the expression.

[tex] \frac{c^{-6} d^4}{c^{-16} d^{14}} [/tex]

A. [tex] c^{10} d^{10} [/tex]
B. [tex] c^2 d^2 [/tex]
C. [tex] \frac{c^{10}}{d^{10}} [/tex]
D. [tex] \frac{d^{-2}}{d^{-2}} [/tex]



Answer :

To simplify the expression [tex]\(\frac{c^{-6} d^4}{c^{-16} d^{14}}\)[/tex], let's follow these steps:

1. Combine the exponents of [tex]\(c\)[/tex]:
- The numerator has [tex]\(c^{-6}\)[/tex].
- The denominator has [tex]\(c^{-16}\)[/tex].
- When dividing powers of the same base, you subtract the exponent in the denominator from the exponent in the numerator.
- [tex]\[ c^{-6 - (-16)} = c^{-6 + 16} = c^{10} \][/tex]

2. Combine the exponents of [tex]\(d\)[/tex]:
- The numerator has [tex]\(d^4\)[/tex].
- The denominator has [tex]\(d^{14}\)[/tex].
- When dividing powers of the same base, you subtract the exponent in the denominator from the exponent in the numerator.
- [tex]\[ d^{4 - 14} = d^{-10} \][/tex]

So the simplified expression is:
[tex]\[ \frac{c^{-6} d^4}{c^{-16} d^{14}} = c^{10} d^{-10} \][/tex]

The answer is not [tex]\(c^{10} d^{10}\)[/tex], [tex]\(c^2 d^2\)[/tex], nor [tex]\(\frac{d^{-2}}{d^{-2}}\)[/tex].

The correct simplified expression is:
[tex]\[ \frac{c^{10}}{d^{10}} \][/tex]
Thus, the proper choice from the given options is:
[tex]\[ \frac{c^{10}}{d^{10}} \][/tex]