Problem

Simplify ((12p^2)/(7d^4))÷((6p^3)/(35d^2))

The given problem is asking to perform the simplification of a complex fraction which involves variables and their exponents. Specifically, it involves dividing one fraction that contains variables raised to powers by another fraction that also contains variables with exponents. The process will likely involve flipping the second fraction (the divisor) to multiply, as is customary in division of fractions, simplifying any common factors, and applying the properties of exponents to simplify the expression to its most reduced form.

12p27d4÷6p335d2

Answer

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Solution:

Step:1

To perform division with fractions, we multiply by the inverse of the divisor. Multiply 12p27d4 by the reciprocal of 6p335d2.

12p27d4×35d26p3

Step:2

Begin simplifying the expression.

Step:2.1

Combine the numerators and denominators.

12p235d27d46p3

Step:2.2

Identify and cancel out common factors between 12 and 6.

Step:2.2.1

Extract the factor of 6 from 12p235d2.

6(2p235d2)7d46p3

Step:2.2.2

Proceed to cancel out the common factors.

Step:2.2.2.1

Extract the factor of 6 from 7d46p3.

6(2p235d2)6(7d4p3)

Step:2.2.2.2

Eliminate the common factor of 6.

6(2p235d2)6(7d4p3)

Step:2.2.2.3

Express the simplified form.

2p235d27d4p3

Step:2.3

Identify and cancel out common factors between p2 and p3.

Step:2.3.1

Extract the factor of p2 from 2p235d2.

p2(235d2)7d4p3

Step:2.3.2

Proceed to cancel out the common factors.

Step:2.3.2.1

Extract the factor of p2 from 7d4p3.

p2(235d2)p2(7d4p)

Step:2.3.2.2

Eliminate the common factor of p2.

p2(235d2)p2(7d4p)

Step:2.3.2.3

Express the simplified form.

235d27d4p

Step:2.4

Identify and cancel out common factors between 35 and 7.

Step:2.4.1

Extract the factor of 7 from 235d2.

7(25d2)7d4p

Step:2.4.2

Proceed to cancel out the common factors.

Step:2.4.2.1

Extract the factor of 7 from 7d4p.

7(25d2)7(d4p)

Step:2.4.2.2

Eliminate the common factor of 7.

7(25d2)7(d4p)

Step:2.4.2.3

Express the simplified form.

25d2d4p

Step:2.5

Identify and cancel out common factors between d2 and d4.

Step:2.5.1

Extract the factor of d2 from 25d2.

d2(25)d4p

Step:2.5.2

Proceed to cancel out the common factors.

Step:2.5.2.1

Extract the factor of d2 from d4p.

d2(25)d2(d2p)

Step:2.5.2.2

Eliminate the common factor of d2.

d2(25)d2(d2p)

Step:2.5.2.3

Express the simplified form.

25d2p

Step:3

Calculate the product of 2 and 5.

10d2p

Knowledge Notes:

To simplify a complex fraction involving division, the following steps are typically taken:

  1. Multiplication by Reciprocal: When dividing by a fraction, the operation is equivalent to multiplying by the reciprocal of that fraction.

  2. Simplification: This involves combining the numerators and denominators into a single fraction, then reducing it by canceling out common factors.

  3. Factorization: Often, it is easier to cancel out common factors when they are factored out of the numerator and denominator.

  4. Cancellation: After factoring, any common factors in the numerator and denominator can be canceled out.

  5. Final Expression: Once all common factors are canceled, the expression should be rewritten in its simplest form.

  6. Multiplication of Constants: If there are any constants left in the numerator, they should be multiplied together to get the final simplified result.

In algebra, these steps are crucial for simplifying expressions and solving equations efficiently. Understanding how to manipulate fractions and factorize terms is fundamental to algebraic operations.

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