Simplify v(((2)(431*10^-3)(9.8))/((1.3)p(10.5*10^-3)^2))

Solution:

Step 1: Perform the initial multiplication.

Step 1.1: Multiply $2$ by $431$ to get $v\frac{862\cdot {10}^{-3}\cdot 9.8}{1.3p(10.5\cdot {10}^{-3}{)}^2}$.

Step 1.2: Multiply $862$ by $9.8$ to obtain $v\frac{8447.6\cdot {10}^{-3}}{1.3p(10.5\cdot {10}^{-3}{)}^2}$.

Step 2: Simplify the numerator.

Step 2.1: Adjust the decimal point in $8447.6$ three places to the left, which gives $v\frac{8.4476\cdot {10}^0}{1.3p(10.5\cdot {10}^{-3}{)}^2}$.

Step 2.2: Convert $8.4476\cdot {10}^0$ to its non-scientific form, resulting in $v\frac{8.4476}{1.3p(10.5\cdot {10}^{-3}{)}^2}$.

Step 3: Calculate the denominator by squaring $10.5\cdot {10}^{-3}$ and multiplying by $1.3$, which simplifies to $v\frac{8.4476}{0.00014332p}$.

Step 4: Factor out common terms.

Step 4.1: Extract $8.4476$ from the numerator to get $v\frac{8.4476(1)}{0.00014332p}$.

Step 4.2: Factor out $0.00014332$ from the denominator, leading to $v\frac{8.4476(1)}{0.00014332(p)}$.

Step 5: Separate the fractions to form $v(\frac{8.4476}{0.00014332}\cdot \frac{1}{p})$.

Step 6: Divide $8.4476$ by $0.00014332$ to simplify the expression to $v\left(\frac{1}{p}\right)$.

Step 7: Apply the commutative property of multiplication to rewrite as $v\frac{1}{p}$.

Step 8: Combine the variables and fractions.

Step 8.1: Merge $v$ and $\frac{1}{p}$ to form $v\frac{}{p}$.

Step 8.2: Finalize the combination to get $\frac{v\cdot }{p}$.

Step 9: Reorder the terms to present the simplified expression as $\frac{v}{p}$.

Knowledge Notes:

1. Multiplication of Scientific Notation: When multiplying numbers in scientific notation, you multiply the base numbers and add the exponents of the powers of ten. For example, $a\cdot {10}^m\times b\cdot {10}^n=(a\times b)\cdot {10}^{m+n}$.

2. Simplifying Numerators and Denominators: To simplify a fraction, you can divide both the numerator and the denominator by the same number (factoring) or perform arithmetic operations to reduce the fraction to its simplest form.

3. Squaring Numbers in Scientific Notation: When you square a number in scientific notation, you square the base number and multiply the exponent by two. For example, $(c\cdot {10}^p{)}^2=c^2\cdot {10}^{2p}$.

4. Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not affect the product. For example, $a\times b=b\times a$.

5. Separating Fractions: When you have a fraction with multiple terms in the numerator or denominator, you can separate them into individual fractions that can be simplified independently, as long as you do not change the overall value of the expression.

6. Division of Decimals: When dividing decimal numbers, you can move the decimal point in both the numerator and the denominator to make the division easier, as long as you move them by the same number of places.

7. Final Expression: The goal of simplification is to rewrite the expression in the simplest form possible, which often involves reducing fractions, factoring out common terms, and applying arithmetic properties.