Final answer:
The distributive property in mathematics allows for multiplication to be distributed across addition or subtraction within brackets, facilitating the simplification of expressions. When multiplying fractions, one should multiply numerators and denominators separately, and the property also helps in clearing fractions by multiplying both sides of an equation by the same number.
Step-by-step explanation:
The Distributive Property
The distributive property is a fundamental concept in algebra which states that multiplication is distributed over addition or subtraction within parentheses. For example, using the distributive property, we can rewrite the expression A(B+C) as AB + AC. This is particularly useful when dealing with more complex expressions that cannot be simplified easily by other means. The approach we take can depend on what is easier to understand or what makes the math more straightforward.
When multiplying fractions, remember to multiply the numerators (top numbers) and the denominators (bottom numbers) separately. The rules for multiplication of signs also apply; two positive numbers yield a positive product, two negative numbers yield a positive product, and numbers with opposite signs yield a negative product. These rules apply similarly to division.
When we need to clear fractions or deal with denominators in an equation, multiplying both sides of the equation by the same non-zero number will not change the equality and makes it easier to solve the equation. This practice ensures that we can transform equations into more manageable forms without altering their solutions.
For example, if we have an equation 1/2x + 3 = 7, we can multiply every term by 2 (the denominator of the fraction) to get x + 6 = 14, which is easier to solve for x.