In the realm of mathematics, expressions involving variables can sometimes be represented in multiple forms. One such example is the expression xy^2/9, which can be expressed in various equivalent forms. Understanding these equivalencies is crucial for algebraic manipulations and problem-solving.
Simplifying xy^2/9
The expression xy^2/9 can be simplified by factoring out any common factors from the numerator and denominator. In this case, we can identify that y is a common factor in both the numerator and denominator:
xy^2/9 = (y * x^1) / (y * 3^2)
Cancelling out the common factor y, we get:
xy^2/9 = x / 9
Therefore, xy^2/9 is equivalent to x/9.
Other Equivalent Expressions
In addition to x/9, there are other expressions that are also equivalent to xy^2/9. These include:
- (x/3) * (y^2/3): This expression is obtained by multiplying the numerator and denominator of xy^2/9 by 3:
xy^2/9 = [(x * 3) * (y^2 * 3)] / [(3 * 3) * (3 * 3)]
Simplifying, we get:
xy^2/9 = (x/3) * (y^2/3)
- (xy)^2 / 9: This expression is obtained by squaring the numerator and denominator of xy^2/9:
xy^2/9 = [(xy) * (xy)] / [(3) * (3)]
Simplifying, we get:
xy^2/9 = (xy)^2 / 9
Applications of Equivalency
Understanding the equivalencies of expressions is essential for various mathematical applications, such as:
- Solving equations: When solving equations involving variables, it can be useful to replace one expression with its equivalent to simplify the equation and find the solution more easily.
- Simplifying complex expressions: Equivalencies can help simplify complex expressions by replacing them with simpler and more manageable forms.
- Proving mathematical identities: Equivalencies play a crucial role in proving mathematical identities, which involve showing that two expressions are equal for all values of the variables.
- Finding common denominators: When adding or subtracting fractions, finding a common denominator is necessary. Equivalents can be used to express fractions with different denominators in terms of a common denominator.
Tables of Equivalencies
For convenience, here are four tables summarizing the equivalencies discussed above:
Table 1: Equivalents of xy^2/9
| Expression | Equivalent Expression |
|---|---|
| xy^2/9 | x/9 |
| xy^2/9 | (x/3) * (y^2/3) |
| xy^2/9 | (xy)^2 / 9 |
Table 2: Simplifying Equivalents
| Expression | Simplified Expression |
|---|---|
| (x/3) * (y^2/3) | x * y^2 / 9 |
| (xy)^2 / 9 | x^2 * y^4 / 81 |
Table 3: Common Denominators
| Expression | Common Denominator |
|---|---|
| x/9 | 9 |
| (x/3) * (y^2/3) | 9 |
| (xy)^2 / 9 | 81 |
Table 4: Proving Identities
| Identity | Equivalent Expressions |
|---|---|
| x^2 / 9 + y^2 / 9 = (x/3)^2 + (y/3)^2 | xy^2/9 = (x/3) * (y^2/3) |
Conclusion
Understanding the equivalencies of expressions is a fundamental skill in mathematics. It allows us to simplify complex expressions, solve equations, prove identities, and find common denominators. The equivalencies discussed in this article for the expression xy^2/9 provide valuable tools for algebraic manipulations and problem-solving.