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.
$\frac{12 p^{2}}{7 d^{4}} \div \frac{6 p^{3}}{35 d^{2}}$
Answer
Solution:
Step:1
To perform division with fractions, we multiply by the inverse of the divisor. Multiply $\frac{12 p^{2}}{7 d^{4}}$ by the reciprocal of $\frac{6 p^{3}}{35 d^{2}}$.
$$\frac{12 p^{2}}{7 d^{4}} \times \frac{35 d^{2}}{6 p^{3}}$$
Step:2
Begin simplifying the expression.
Step:2.1
Combine the numerators and denominators.
$$\frac{12 p^{2} \cdot 35 d^{2}}{7 d^{4} \cdot 6 p^{3}}$$
Step:2.2
Identify and cancel out common factors between 12 and 6.
Step:2.2.1
Extract the factor of 6 from $12 p^{2} \cdot 35 d^{2}$.
$$\frac{6 \cdot (2 p^{2} \cdot 35 d^{2})}{7 d^{4} \cdot 6 p^{3}}$$
Step:2.2.2
Proceed to cancel out the common factors.
Step:2.2.2.1
Extract the factor of 6 from $7 d^{4} \cdot 6 p^{3}$.
$$\frac{6 \cdot (2 p^{2} \cdot 35 d^{2})}{6 \cdot (7 d^{4} \cdot p^{3})}$$
Step:2.2.2.2
Eliminate the common factor of 6.
$$\frac{\cancel{6} \cdot (2 p^{2} \cdot 35 d^{2})}{\cancel{6} \cdot (7 d^{4} \cdot p^{3})}$$
Step:2.2.2.3
Express the simplified form.
$$\frac{2 p^{2} \cdot 35 d^{2}}{7 d^{4} \cdot p^{3}}$$
Step:2.3
Identify and cancel out common factors between $p^{2}$ and $p^{3}$.
Step:2.3.1
Extract the factor of $p^{2}$ from $2 p^{2} \cdot 35 d^{2}$.
$$\frac{p^{2} \cdot (2 \cdot 35 d^{2})}{7 d^{4} \cdot p^{3}}$$
Step:2.3.2
Proceed to cancel out the common factors.
Step:2.3.2.1
Extract the factor of $p^{2}$ from $7 d^{4} \cdot p^{3}$.
$$\frac{p^{2} \cdot (2 \cdot 35 d^{2})}{p^{2} \cdot (7 d^{4} \cdot p)}$$
Step:2.3.2.2
Eliminate the common factor of $p^{2}$.
$$\frac{\cancel{p^{2}} \cdot (2 \cdot 35 d^{2})}{\cancel{p^{2}} \cdot (7 d^{4} \cdot p)}$$
Step:2.3.2.3
Express the simplified form.
$$\frac{2 \cdot 35 d^{2}}{7 d^{4} \cdot p}$$
Step:2.4
Identify and cancel out common factors between 35 and 7.
Step:2.4.1
Extract the factor of 7 from $2 \cdot 35 d^{2}$.
$$\frac{7 \cdot (2 \cdot 5 d^{2})}{7 d^{4} \cdot p}$$
Step:2.4.2
Proceed to cancel out the common factors.
Step:2.4.2.1
Extract the factor of 7 from $7 d^{4} \cdot p$.
$$\frac{7 \cdot (2 \cdot 5 d^{2})}{7 \cdot (d^{4} \cdot p)}$$
Step:2.4.2.2
Eliminate the common factor of 7.
$$\frac{\cancel{7} \cdot (2 \cdot 5 d^{2})}{\cancel{7} \cdot (d^{4} \cdot p)}$$
Step:2.4.2.3
Express the simplified form.
$$\frac{2 \cdot 5 d^{2}}{d^{4} \cdot p}$$
Step:2.5
Identify and cancel out common factors between $d^{2}$ and $d^{4}$.
Step:2.5.1
Extract the factor of $d^{2}$ from $2 \cdot 5 d^{2}$.
$$\frac{d^{2} \cdot (2 \cdot 5)}{d^{4} \cdot p}$$
Step:2.5.2
Proceed to cancel out the common factors.
Step:2.5.2.1
Extract the factor of $d^{2}$ from $d^{4} \cdot p$.
$$\frac{d^{2} \cdot (2 \cdot 5)}{d^{2} \cdot (d^{2} \cdot p)}$$
Step:2.5.2.2
Eliminate the common factor of $d^{2}$.
$$\frac{\cancel{d^{2}} \cdot (2 \cdot 5)}{\cancel{d^{2}} \cdot (d^{2} \cdot p)}$$
Step:2.5.2.3
Express the simplified form.
$$\frac{2 \cdot 5}{d^{2} \cdot p}$$
Step:3
Calculate the product of 2 and 5.
$$\frac{10}{d^{2} p}$$
Knowledge Notes:
To simplify a complex fraction involving division, the following steps are typically taken:
Multiplication by Reciprocal: When dividing by a fraction, the operation is equivalent to multiplying by the reciprocal of that fraction.
Simplification: This involves combining the numerators and denominators into a single fraction, then reducing it by canceling out common factors.
Factorization: Often, it is easier to cancel out common factors when they are factored out of the numerator and denominator.
Cancellation: After factoring, any common factors in the numerator and denominator can be canceled out.
Final Expression: Once all common factors are canceled, the expression should be rewritten in its simplest form.
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.
