Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To determine which expression is equivalent to [tex]\(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\)[/tex], we start by understanding that division of fractions is equivalent to multiplying by the reciprocal of the second fraction.
Given expression:
[tex]\[ \frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)} \][/tex]
First rewrite the division as a multiplication by the reciprocal:
[tex]\[ \frac{c^2-4}{c+3} \times \frac{3(c^2-9)}{c+2} \][/tex]
Next, we'll simplify the components of the expression.
1. Factorize [tex]\(c^2 - 4\)[/tex] and [tex]\(c^2 - 9\)[/tex] using the difference of squares:
[tex]\[ c^2 - 4 = (c + 2)(c - 2) \][/tex]
[tex]\[ c^2 - 9 = (c + 3)(c - 3) \][/tex]
2. Substitute the factored forms into the expression:
[tex]\[ \frac{(c + 2)(c - 2)}{c + 3} \times \frac{3((c + 3)(c - 3))}{c + 2} \][/tex]
3. Simplify the expression by canceling out the common terms:
[tex]\[ = \frac{(c + 2)(c - 2)}{c + 3} \times \frac{3(c + 3)(c - 3)}{c + 2} \][/tex]
Notice that [tex]\(c + 2\)[/tex] in the numerator and denominator can be canceled:
[tex]\[ = \frac{(c - 2)}{c + 3} \times 3(c + 3)(c - 3) \][/tex]
Now cancel [tex]\(c + 3\)[/tex] in the numerator and denominator:
[tex]\[ = \frac{c - 2}{1} \times 3(c - 3) \][/tex]
This simplifies further to:
[tex]\[ = 3(c - 2) / (c - 3) \][/tex]
Thus, the simplified expression is:
[tex]\[ 3 \cdot \frac{(c - 2)}{(c - 3)} \][/tex]
So, the expression equivalent to [tex]\(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\)[/tex] is:
[tex]\[ \boxed{\frac{c^2-4}{c+3} \cdot \frac{3 \left( c^2-9 \right)}{c+2}} \][/tex]
Rewrite this expression for clarity as:
[tex]\[ 3 \cdot \frac{(c-2)}{(c-3)} \][/tex]
Given expression:
[tex]\[ \frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)} \][/tex]
First rewrite the division as a multiplication by the reciprocal:
[tex]\[ \frac{c^2-4}{c+3} \times \frac{3(c^2-9)}{c+2} \][/tex]
Next, we'll simplify the components of the expression.
1. Factorize [tex]\(c^2 - 4\)[/tex] and [tex]\(c^2 - 9\)[/tex] using the difference of squares:
[tex]\[ c^2 - 4 = (c + 2)(c - 2) \][/tex]
[tex]\[ c^2 - 9 = (c + 3)(c - 3) \][/tex]
2. Substitute the factored forms into the expression:
[tex]\[ \frac{(c + 2)(c - 2)}{c + 3} \times \frac{3((c + 3)(c - 3))}{c + 2} \][/tex]
3. Simplify the expression by canceling out the common terms:
[tex]\[ = \frac{(c + 2)(c - 2)}{c + 3} \times \frac{3(c + 3)(c - 3)}{c + 2} \][/tex]
Notice that [tex]\(c + 2\)[/tex] in the numerator and denominator can be canceled:
[tex]\[ = \frac{(c - 2)}{c + 3} \times 3(c + 3)(c - 3) \][/tex]
Now cancel [tex]\(c + 3\)[/tex] in the numerator and denominator:
[tex]\[ = \frac{c - 2}{1} \times 3(c - 3) \][/tex]
This simplifies further to:
[tex]\[ = 3(c - 2) / (c - 3) \][/tex]
Thus, the simplified expression is:
[tex]\[ 3 \cdot \frac{(c - 2)}{(c - 3)} \][/tex]
So, the expression equivalent to [tex]\(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\)[/tex] is:
[tex]\[ \boxed{\frac{c^2-4}{c+3} \cdot \frac{3 \left( c^2-9 \right)}{c+2}} \][/tex]
Rewrite this expression for clarity as:
[tex]\[ 3 \cdot \frac{(c-2)}{(c-3)} \][/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.