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Which expression is equivalent to [tex]\(\frac{9 x^5 y^{16}}{45 x^5 y^4}\)[/tex]?

A. [tex]\(\frac{y^{12}}{5}\)[/tex]

B. [tex]\(\frac{y^{12}}{36}\)[/tex]

C. [tex]\(\frac{x y^4}{5}\)[/tex]

D. [tex]\(\frac{x y^4}{36}\)[/tex]


Sagot :

To simplify the given expression [tex]\(\frac{9 x^5 y^{16}}{45 x^5 y^4}\)[/tex], we need to break it down step-by-step:

1. Simplify the coefficients:

The coefficients in the expression are 9 in the numerator and 45 in the denominator. Simplify the fraction [tex]\(\frac{9}{45}\)[/tex]:
[tex]\[ \frac{9}{45} = \frac{9 \div 9}{45 \div 9} = \frac{1}{5} \][/tex]

2. Simplify the [tex]\(x\)[/tex] terms:

The [tex]\(x\)[/tex] terms are [tex]\(x^5\)[/tex] in both the numerator and the denominator. Therefore, we can cancel these terms, as [tex]\(x^5 / x^5 = x^{5-5} = x^0 = 1\)[/tex]:
[tex]\[ \frac{x^5}{x^5} = 1 \][/tex]

3. Simplify the [tex]\(y\)[/tex] terms:

The [tex]\(y\)[/tex] terms are [tex]\(y^{16}\)[/tex] in the numerator and [tex]\(y^4\)[/tex] in the denominator. Use the properties of exponents to simplify:
[tex]\[ \frac{y^{16}}{y^4} = y^{16-4} = y^{12} \][/tex]

Putting all the simplified parts together, we have:
[tex]\[ \frac{9 x^5 y^{16}}{45 x^5 y^4} = \frac{1}{5} \times y^{12} = \frac{y^{12}}{5} \][/tex]

Therefore, the expression equivalent to [tex]\(\frac{9 x^5 y^{16}}{45 x^5 y^4}\)[/tex] is:
[tex]\[ \boxed{\frac{y^{12}}{5}} \][/tex]
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