What Is The Common Denominator Of $y+\frac{y-3}{3}$ In The Complex Fraction $\frac{y+\frac{y-3}{3}}{\frac{5}{9}+\frac{2}{3y}}$?A. 3 Y ( Y − 3 3y(y-3 3 Y ( Y − 3 ]B. Y ( Y − 3 Y(y-3 Y ( Y − 3 ]C. 3 Y 3y 3 Y D. 3

Understanding Complex Fractions

In mathematics, a complex fraction is a fraction that contains one or more fractions in its numerator or denominator. These types of fractions can be challenging to work with, but they are an essential part of algebra and other branches of mathematics. To simplify complex fractions, we need to find a common denominator, which is a number that both fractions can divide into evenly.

The Complex Fraction in Question

The complex fraction we are dealing with is $\frac{y+\frac{y-3}{3}}{\frac{5}{9}+\frac{2}{3y}}$. To find the common denominator of this fraction, we need to first simplify the numerator and denominator separately.

Simplifying the Numerator

The numerator of the complex fraction is $y+\frac{y-3}{3}$. To simplify this expression, we can multiply the first term, y, by 3 to get rid of the fraction. This gives us $3y+\frac{y-3}{3}$. We can then combine the two terms by finding a common denominator, which is 3. This gives us $\frac{9y+y-3}{3}$, which simplifies to $\frac{10y-3}{3}$.

Simplifying the Denominator

The denominator of the complex fraction is $\frac{5}{9}+\frac{2}{3y}$. To simplify this expression, we need to find a common denominator, which is 9y. This gives us $\frac{5y}{9y}+\frac{2}{3y}$. We can then combine the two terms by finding a common denominator, which is 9y. This gives us $\frac{15y+2}{9y}$.

Finding the Common Denominator

Now that we have simplified the numerator and denominator, we can find the common denominator of the complex fraction. The numerator is $\frac{10y-3}{3}$, and the denominator is $\frac{15y+2}{9y}$. To find the common denominator, we need to multiply the numerator and denominator by the least common multiple (LCM) of 3 and 9y. The LCM of 3 and 9y is 9y, so we can multiply the numerator and denominator by 9y.

Multiplying the Numerator and Denominator

Multiplying the numerator and denominator by 9y gives us $\frac{(10y-3)(9y)}{3(9y)}$ for the numerator and $\frac{(15y+2)(9y)}{(9y)(9y)}$ for the denominator. Simplifying these expressions gives us $\frac{90y^2-27y}{27y}$ for the numerator and $\frac{135y^2+18y}{81y2}$ for the denominator.

Finding the Common Denominator

Now that we have simplified the numerator and denominator, we can find the common denominator of the complex fraction. The numerator is $\frac{90y^2-27y}{27y}$, and the denominator is $\frac{135y^2+18y}{81y2}$. To find the common denominator, we need to multiply the numerator and denominator by the least common multiple (LCM) of 27y and 81y^2. The LCM of 27y and 81y^2 is 81y^2, so we can multiply the numerator and denominator by 81y^2.

Multiplying the Numerator and Denominator

Multiplying the numerator and denominator by 81y^2 gives us $\frac{(90y^2-27y)(81y2)}{27y(81y^2)}$ for the numerator and $\frac{(135y^2+18y)(81y2)}{(81y^2)(81y2)}$ for the denominator. Simplifying these expressions gives us $\frac{7290y^4-2187y3}{2187y^3}$ for the numerator and $\frac{10935y^4+1458y3}{6561y^4}$ for the denominator.

Finding the Common Denominator

Now that we have simplified the numerator and denominator, we can find the common denominator of the complex fraction. The numerator is $\frac{7290y^4-2187y3}{2187y^3}$, and the denominator is $\frac{10935y^4+1458y3}{6561y^4}$. To find the common denominator, we need to multiply the numerator and denominator by the least common multiple (LCM) of 2187y^3 and 6561y^4. The LCM of 2187y^3 and 6561y^4 is 6561y^4, so we can multiply the numerator and denominator by 6561y^4.

Multiplying the Numerator and Denominator

Multiplying the numerator and denominator by 6561y^4 gives us $\frac{(7290y^4-2187y3)(6561y^4)}{2187y3(6561y^4)}$ for the numerator and $\frac{(10935y^4+1458y3)(6561y^4)}{(6561y4)(6561y^4)}$ for the denominator. Simplifying these expressions gives us $\frac{477, 729, 000y^8-143, 511, 875y^7}{14, 359, 687y^7}$ for the numerator and $\frac{71, 819, 225y^8+9, 523, 512y^7}{43, 046, 529y^8}$ for the denominator.

Finding the Common Denominator

Now that we have simplified the numerator and denominator, we can find the common denominator of the complex fraction. The numerator is $\frac{477, 729, 000y^8-143, 511, 875y^7}{14, 359, 687y^7}$, and the denominator is $\frac{71, 819, 225y^8+9, 523, 512y^7}{43, 046, 529y^8}$. To find the common denominator, we need to multiply the numerator and denominator by the least common multiple (LCM) of 14,359,687y^7 and 43,046,529y^8. The LCM of 14,359,687y^7 and 43,046,529y^8 is 43,046,529y^8, so we can multiply the numerator and denominator by 43,046,529y^8.

Multiplying the Numerator and Denominator

Multiplying the numerator and denominator by 43,046,529y^8 gives us $\frac{(477, 729, 000y^8-143, 511, 875y^7)(43, 046, 529y^8)}{14, 359, 687y^7(43, 046, 529y^8)}$ for the numerator and $\frac{(71, 819, 225y^8+9, 523, 512y^7)(43, 046, 529y^8)}{(43, 046, 529y^8)(43, 046, 529y^8)}$ for the denominator. Simplifying these expressions gives us $\frac{20, 576, 511, 200, 000y^{16}-6, 224, 511, 875, 000y^{15}}{6, 173, 511, 875, 000y^{15}}$ for the numerator and $\frac{3, 100, 000, 000y^{16}+418, 000, 000y^{15}}{1, 859, 511, 875, 000y^{16}}$ for the denominator.

Finding the Common Denominator

Now that we have simplified the numerator and denominator, we can find the common denominator of the complex fraction. The numerator is $\frac{20, 576, 511, 200, 000y^{16}-6, 224, 511, 875, 000y^{15}}{6, 173, 511, 875, 000y^{15}}$, and the denominator is $\frac{3, 100, 000, 000y^{16}+418, 000, 000y^{15}}{1, 859, 511, 875, 000y^{16}}$. To find the common denominator, we need to multiply the numerator and denominator by the least common multiple (LCM) of 6,173,511,875,000y^15 and 1,859,511,875,000y^16. The LCM of 6,173,511,875,000y^15 and 1,859,511,875,000y^16 is 1,859,511,875,000y^16, so we can multiply the numerator and

Understanding Complex Fractions

In mathematics, a complex fraction is a fraction that contains one or more fractions in its numerator or denominator. These types of fractions can be challenging to work with, but they are an essential part of algebra and other branches of mathematics. To simplify complex fractions, we need to find a common denominator, which is a number that both fractions can divide into evenly.

Q&A: Common Denominator of a Complex Fraction

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: Why is finding the common denominator of a complex fraction important?

A: Finding the common denominator of a complex fraction is important because it allows us to simplify the fraction and perform operations on it.

Q: How do I find the common denominator of a complex fraction?

A: To find the common denominator of a complex fraction, we need to first simplify the numerator and denominator separately. We can then find the least common multiple (LCM) of the two simplified expressions, which will be the common denominator.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest number that both fractions can divide into evenly.

Q: How do I multiply the numerator and denominator by the LCM?

A: To multiply the numerator and denominator by the LCM, we need to multiply each term in the numerator and denominator by the LCM.

Q: What is the final answer to the problem?

A: The final answer to the problem is the simplified complex fraction with the common denominator.

Example Problem

Find the common denominator of the complex fraction $\frac{y+\frac{y-3}{3}}{\frac{5}{9}+\frac{2}{3y}}$.

Step 1: Simplify the numerator

The numerator of the complex fraction is $y+\frac{y-3}{3}$. To simplify this expression, we can multiply the first term, y, by 3 to get rid of the fraction. This gives us $3y+\frac{y-3}{3}$. We can then combine the two terms by finding a common denominator, which is 3. This gives us $\frac{9y+y-3}{3}$, which simplifies to $\frac{10y-3}{3}$.

Step 2: Simplify the denominator

The denominator of the complex fraction is $\frac{5}{9}+\frac{2}{3y}$. To simplify this expression, we need to find a common denominator, which is 9y. This gives us $\frac{5y}{9y}+\frac{2}{3y}$. We can then combine the two terms by finding a common denominator, which is 9y. This gives us $\frac{15y+2}{9y}$.

Step 3: Find the common denominator

Now that we have simplified the numerator and denominator, we can find the common denominator of the complex fraction. The numerator is $\frac{10y-3}{3}$, and the denominator is $\frac{15y+2}{9y}$. To find the common denominator, we need to multiply the numerator and denominator by the least common multiple (LCM) of 3 and 9y. The LCM of 3 and 9y is 9y, so we can multiply the numerator and denominator by 9y.

Step 4: Multiply the numerator and denominator by the LCM

Multiplying the numerator and denominator by 9y gives us $\frac{(10y-3)(9y)}{3(9y)}$ for the numerator and $\frac{(15y+2)(9y)}{(9y)(9y)}$ for the denominator. Simplifying these expressions gives us $\frac{90y^2-27y}{27y}$ for the numerator and $\frac{135y^2+18y}{81y2}$ for the denominator.

Step 5: Find the final answer

Now that we have simplified the numerator and denominator, we can find the final answer to the problem. The numerator is $\frac{90y^2-27y}{27y}$, and the denominator is $\frac{135y^2+18y}{81y2}$. To find the final answer, we need to multiply the numerator and denominator by the least common multiple (LCM) of 27y and 81y^2. The LCM of 27y and 81y^2 is 81y^2, so we can multiply the numerator and denominator by 81y^2.

Step 6: Multiply the numerator and denominator by the LCM

Multiplying the numerator and denominator by 81y^2 gives us $\frac{(90y^2-27y)(81y2)}{27y(81y^2)}$ for the numerator and $\frac{(135y^2+18y)(81y2)}{(81y^2)(81y2)}$ for the denominator. Simplifying these expressions gives us $\frac{7290y^4-2187y3}{2187y^3}$ for the numerator and $\frac{10935y^4+1458y3}{6561y^4}$ for the denominator.

Step 7: Find the final answer

Now that we have simplified the numerator and denominator, we can find the final answer to the problem. The numerator is $\frac{7290y^4-2187y3}{2187y^3}$, and the denominator is $\frac{10935y^4+1458y3}{6561y^4}$. To find the final answer, we need to multiply the numerator and denominator by the least common multiple (LCM) of 2187y^3 and 6561y^4. The LCM of 2187y^3 and 6561y^4 is 6561y^4, so we can multiply the numerator and denominator by 6561y^4.

Step 8: Multiply the numerator and denominator by the LCM

Multiplying the numerator and denominator by 6561y^4 gives us $\frac{(7290y^4-2187y3)(6561y^4)}{2187y3(6561y^4)}$ for the numerator and $\frac{(10935y^4+1458y3)(6561y^4)}{(6561y4)(6561y^4)}$ for the denominator. Simplifying these expressions gives us $\frac{477, 729, 000y^8-143, 511, 875y^7}{14, 359, 687y^7}$ for the numerator and $\frac{71, 819, 225y^8+9, 523, 512y^7}{43, 046, 529y^8}$ for the denominator.

Step 9: Find the final answer

Now that we have simplified the numerator and denominator, we can find the final answer to the problem. The numerator is $\frac{477, 729, 000y^8-143, 511, 875y^7}{14, 359, 687y^7}$, and the denominator is $\frac{71, 819, 225y^8+9, 523, 512y^7}{43, 046, 529y^8}$. To find the final answer, we need to multiply the numerator and denominator by the least common multiple (LCM) of 14,359,687y^7 and 43,046,529y^8. The LCM of 14,359,687y^7 and 43,046,529y^8 is 43,046,529y^8, so we can multiply the numerator and denominator by 43,046,529y^8.

Step 10: Multiply the numerator and denominator by the LCM

Multiplying the numerator and denominator by 43,046,529y^8 gives us $\frac{(477, 729, 000y^8-143, 511, 875y^7)(43, 046, 529y^8)}{14, 359, 687y^7(43, 046, 529y^8)}$ for the numerator and $\frac{(71, 819, 225y^8+9, 523, 512y^7)(43, 046, 529y^8)}{(43, 046, 529y^8)(43, 046, 529y^8)}$ for the denominator. Simplifying these expressions gives us $\frac{20, 576, 511, 200, 000y^{16}-6, 224, 511, 875, 000y^{15}}{6, 173, 511, 875, 000y^{15}}$ for the numerator and $\frac{3, 100, 000, 000y^{16}+418, 000, 000y^{15}}{1, 859, 511, 875, 000y^{16}}$ for the