Evaluate (1+2/3)*(1+2/5)*(1+2/7)*(1+2/9)

Solution:

Step 1: Simplify the given expression.

Step 1.1

Express $1$ as a fraction with a denominator that matches the adjacent term. $(\frac{3}{3}+\frac{2}{3})\cdot (1+\frac{2}{5})\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 1.2

Combine the numerators over the shared denominator. $\frac{3+2}{3}\cdot (1+\frac{2}{5})\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 1.3

Perform the addition in the numerator. $\frac{5}{3}\cdot (1+\frac{2}{5})\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 1.4

Again, express $1$ as a fraction with a suitable denominator. $\frac{5}{3}\cdot (\frac{5}{5}+\frac{2}{5})\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 1.5

Combine the numerators over the common denominator. $\frac{5}{3}\cdot \frac{5+2}{5}\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 1.6

Sum the numerators. $\frac{5}{3}\cdot \frac{7}{5}\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 2: Eliminate the common factor of $5$.

Step 2.1

Remove the common factor. $\frac{\cancel{5}}{3}\cdot \frac{7}{\cancel{5}}\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 2.2

Simplify the expression. $\frac{1}{3}\cdot 7\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 3: Combine $\frac{1}{3}$ and $7$. $\frac{7}{3}\cdot (1+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 4: Simplify further.

Step 4.1

Express $1$ as a fraction. $\frac{7}{3}\cdot (\frac{7}{7}+\frac{2}{7})\cdot (1+\frac{2}{9})$

Step 4.2

Combine the numerators. $\frac{7}{3}\cdot \frac{7+2}{7}\cdot (1+\frac{2}{9})$

Step 4.3

Add the numerators. $\frac{7}{3}\cdot \frac{9}{7}\cdot (1+\frac{2}{9})$

Step 5: Cancel the common factor of $7$.

Step 5.1

Remove the common factor. $\frac{\cancel{7}}{3}\cdot \frac{9}{\cancel{7}}\cdot (1+\frac{2}{9})$

Step 5.2

Simplify the expression. $\frac{1}{3}\cdot 9\cdot (1+\frac{2}{9})$

Step 6: Eliminate the common factor of $3$.

Step 6.1

Extract $3$ from $9$. $\frac{1}{3}\cdot (3\cdot 3)\cdot (1+\frac{2}{9})$

Step 6.2

Cancel the common factor. $\frac{1}{\cancel{3}}\cdot (\cancel{3}\cdot 3)\cdot (1+\frac{2}{9})$

Step 6.3

Rewrite the simplified expression. $3\cdot (1+\frac{2}{9})$

Step 7: Further simplification.

Step 7.1

Convert $1$ to a fraction. $3\cdot (\frac{9}{9}+\frac{2}{9})$

Step 7.2

Combine the numerators. $3\cdot \frac{9+2}{9}$

Step 7.3

Sum the numerators. $3\cdot \frac{11}{9}$

Step 8: Cancel the common factor of $3$.

Step 8.1

Factor $3$ from $9$. $3\cdot \frac{11}{3\cdot 3}$

Step 8.2

Remove the common factor. $\cancel{3}\cdot \frac{11}{\cancel{3}\cdot 3}$

Step 8.3

Present the simplified expression. $\frac{11}{3}$

Step 9: Represent the result in various forms.

Exact Form: $\frac{11}{3}$ Decimal Form: $3.\bar{6}$ Mixed Number Form: $3\frac{2}{3}$

Knowledge Notes:

To solve the given problem, we used several mathematical concepts and techniques:

1. Fraction Addition: When adding fractions with the same denominator, we simply add the numerators and keep the denominator the same.

2. Simplification: We simplified the expression by combining like terms and reducing fractions when possible.

3. Cancellation: When a number appears in both the numerator and the denominator, it can be canceled out, which is essentially dividing both by that number.

4. Multiplication of Fractions: To multiply fractions, we multiply the numerators together and the denominators together.

5. Conversion to Mixed Number: A fraction greater than one can be expressed as a mixed number, which is a whole number plus a proper fraction.

These concepts are fundamental in algebra and are often used in simplifying expressions and solving equations.