Problem

Simplify (-10d^4e^3f^5)/(15d^2e^2f)

The given problem is asking to simplify a mathematical expression that involves the division of two algebraic terms, each consisting of a numerical coefficient and variables raised to certain powers. The operation to be performed is to simplify the expression by dividing the numerical coefficients and subtract the exponents of the corresponding variables with like bases in the numerator and the denominator according to the laws of exponents. This involves reducing the fraction to its simplest form by canceling out common factors and applying the exponent rules for division.

10d4e3f515d2e2f

Answer

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Solution:

Step 1:

Identify and simplify the common numerical factor between 10 and 15.

Step 1.1:

Extract the factor of 5 from 10d4e3f5 to get 5(2d4e3f5)15d2e2f.

Step 1.2:

Proceed to reduce the common numerical factors.

Step 1.2.1:

Extract the factor of 5 from 15d2e2f to obtain 5(2d4e3f5)5(3d2e2f).

Step 1.2.2:

Eliminate the common factor of 5 resulting in 5(2d4e3f5)5(3d2e2f).

Step 1.2.3:

Restate the simplified expression as 2d4e3f53d2e2f.

Step 2:

Reduce the common variable factor between d4 and d2.

Step 2.1:

Factor out d2 from 2d4e3f5 to get d2(2d2e3f5)3d2e2f.

Step 2.2:

Proceed to cancel out the common variable factors.

Step 2.2.1:

Factor out d2 from 3d2e2f to obtain d2(2d2e3f5)d2(3e2f).

Step 2.2.2:

Eliminate the common factor of d2 resulting in d2(2d2e3f5)d2(3e2f).

Step 2.2.3:

Restate the simplified expression as 2d2e3f53e2f.

Step 3:

Reduce the common variable factor between e3 and e2.

Step 3.1:

Factor out e2 from 2d2e3f5 to get e2(2d2ef5)3e2f.

Step 3.2:

Proceed to cancel out the common variable factors.

Step 3.2.1:

Factor out e2 from 3e2f to obtain e2(2d2ef5)e2(3f).

Step 3.2.2:

Eliminate the common factor of e2 resulting in e2(2d2ef5)e2(3f).

Step 3.2.3:

Restate the simplified expression as 2d2ef53f.

Step 4:

Reduce the common variable factor between f5 and f.

Step 4.1:

Factor out f from 2d2ef5 to get f(2d2ef4)3f.

Step 4.2:

Proceed to cancel out the common variable factors.

Step 4.2.1:

Factor out f from 3f to obtain f(2d2ef4)f3.

Step 4.2.2:

Eliminate the common factor of f resulting in f(2d2ef4)f3.

Step 4.2.3:

Restate the simplified expression as 2d2ef43.

Step 5:

Position the negative sign in front of the fraction to finalize the expression as 2d2ef43.

Knowledge Notes:

The problem involves simplifying a rational expression by canceling out common factors in the numerator and the denominator. The process requires understanding of the following concepts:

  1. Factorization: The process of breaking down numbers or expressions into their constituent factors, which when multiplied together give the original number or expression.

  2. Common Factors: These are factors that are the same in both the numerator and the denominator of a fraction. They can be numerical or variable factors.

  3. Cancellation: When a factor appears in both the numerator and the denominator, it can be 'cancelled out' or reduced to 1, simplifying the expression.

  4. Exponent Rules: When variables have exponents, the rules of exponents apply. For example, am/an=amn when m>n.

  5. Negative Signs: A negative sign can be moved in front of a fraction or to the numerator or denominator, but it must be accounted for in the final expression.

  6. Simplifying Rational Expressions: The process of reducing expressions to their simplest form by canceling common factors.

  7. LaTeX Typesetting: A typesetting system that is widely used for mathematical and scientific documents, due to its powerful handling of formulas and bibliographies. In this context, LaTeX is used to format mathematical expressions for clarity and precision.

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