## Unraveling the Mystery: Which Expression Equals (st)(6)?

**Wondering how to express (st)(6) in a different way?** **The answer lies in understanding the fundamental properties of multiplication.** Let's dive into the world of algebraic expressions and explore the equivalent representations of (st)(6).

**Why is this important?** Understanding equivalent expressions is crucial in algebra, allowing us to simplify equations, solve problems efficiently, and grasp the underlying structure of mathematical relationships.

**Our analysis:** We meticulously explored the properties of multiplication, focusing on the commutative and associative laws, to determine the expressions equivalent to (st)(6). We've also considered the role of parentheses in influencing the order of operations.

**Key Takeaways from our exploration:**

Equivalent Expression | Description |
---|---|

(6)(st) | Commutative Property: Multiplication is commutative, meaning the order of factors doesn't affect the product. |

6st | Associative Property: Multiplication is associative, allowing us to group factors in different ways without altering the result. |

s(6t) | Distributive Property: Multiplication distributes over addition, but in this case, we use the fact that multiplication is associative to rearrange the factors. |

**Transition:** Now, let's delve into each of these equivalent expressions in detail.

### (st)(6)

**Introduction:** This is our starting point, the original expression we're analyzing.

**Key Aspects:**

**Parentheses:**The parentheses indicate the product of "st" and 6.**Variables:**"s" and "t" represent unknown quantities or variables.

**Discussion:** The expression (st)(6) is a product of three factors: "s," "t," and 6. The parentheses highlight the product of "st" before multiplying by 6.

### (6)(st)

**Introduction:** This expression utilizes the commutative property of multiplication.

**Facets:**

**Commutativity:**Multiplication is commutative, meaning that the order of factors doesn't change the product.**Example:**2 x 3 = 3 x 2. Both expressions equal 6.**Application:**We swap the positions of (st) and 6.

**Summary:** (6)(st) is equivalent to (st)(6) because the order of multiplication doesn't impact the result.

### 6st

**Introduction:** This expression leverages the associative property of multiplication.

**Facets:**

**Associativity:**Multiplication is associative, allowing us to group factors in different ways without affecting the product.**Example:**(2 x 3) x 4 = 2 x (3 x 4). Both expressions equal 24.**Application:**We can remove the parentheses because multiplication is associative.

**Summary:** 6st is equivalent to (st)(6) as the grouping of factors doesn't alter the product.

### s(6t)

**Introduction:** This expression demonstrates the distributive property in a slightly different context.

**Facets:**

**Distributive Property:**Multiplication distributes over addition. However, since we're dealing with a product (st)(6), the associative property is utilized to rearrange the factors.**Example:**2(x + y) = 2x + 2y.**Application:**We group 6 and "t" together, recognizing that multiplication is associative.

**Summary:** s(6t) is equivalent to (st)(6) because of the flexibility provided by the associative property of multiplication.

### FAQ

**Introduction:** Let's address some common queries about equivalent expressions.

**Questions:**

**Q: What happens if "s" and "t" are numbers?****A:**If "s" and "t" are numbers, you can directly perform the multiplication to find the numerical value of the expression.**Q: Can I use these equivalent expressions in equations?****A:**Absolutely! Equivalent expressions can be substituted into equations without changing their solutions.**Q: Are there other equivalent expressions?****A: ** While the above expressions are the most common, you can potentially find other equivalent expressions using algebraic manipulations.

**Summary:** Understanding the properties of multiplication is crucial for manipulating and simplifying algebraic expressions.

**Transition:** Let's explore some tips for working with equivalent expressions.

### Tips for Working with Equivalent Expressions

**Introduction:** Here are some useful tips for navigating the world of equivalent expressions:

**Tips:**

**Remember the properties:**Keep in mind the commutative, associative, and distributive properties of multiplication.**Use parentheses wisely:**Parentheses guide the order of operations.**Simplify where possible:**Combine terms and eliminate unnecessary parentheses.**Check your work:**Always verify that the expressions you derive are indeed equivalent to the original.

**Summary:** Utilizing these tips will enhance your ability to simplify and manipulate expressions, making it easier to solve problems and understand algebraic relationships.

### Summary

This exploration has illuminated the various ways to express (st)(6) through equivalent expressions. We've seen how the commutative, associative, and distributive properties of multiplication provide flexibility in rearranging factors and maintaining the same product. Remember, understanding equivalent expressions is fundamental to mastering algebraic manipulations.

### Closing Message

The journey into the world of equivalent expressions reveals the elegance and power of algebraic properties. As you continue your mathematical endeavors, embrace these tools, and you'll discover the beauty and logic that underlies the structure of mathematics.