Problem 6 - Elementary Algebra

Question

Perform each division. $$ \frac{8 m^{2} n}{3 a^{5} b^{2}} \div \frac{2 m}{15 a^{7} b^{2}} $$

Step-by-Step Solution

Answer

Question: Perform the following division: $$ \frac{8 m^{2} n}{3 a^{5} b^{2}} \div\frac{2m}{15a^{7}b^{2}} $$ Answer: After performing the division as explained in the step-by-step solution, the final answer is: $$ 20 m n a^{2} $$

Step 1: Convert the division to multiplication by finding the reciprocal of the second fraction

To convert the given division to multiplication, we need to find the reciprocal of the second fraction (flip the numerator and denominator) and multiply it with the first fraction. The reciprocal of \(\frac{2m}{15a^{7}b^{2}}\) is \(\frac{15a^{7}b^{2}}{2m}\). Now, we have: $$ \frac{8 m^{2} n}{3 a^{5} b^{2}} \times \frac{15 a^{7} b^{2}}{2 m} $$

Step 2: Multiply numerators with numerators and denominators with denominators

Next, we multiply the numerators together and the denominators together: $$ \frac{8 m^{2} n \times 15 a^{7} b^{2}}{3 a^{5} b^{2} \times 2 m} $$

Step 3: Simplify by canceling out common factors

In this step, we simplify the expression by canceling out the common factors in the numerator and denominator: $$ \frac{8 \cancel{m^{2}} n \times 15 a^{7} \cancel{b^{2}}}{3 a^{5} \cancel{b^{2}} \times 2 \cancel{m}} $$ After canceling common factors, we are left with: $$ \frac{8 m n \times 15 a^{7}}{3 a^{5} \times 2} $$

Step 4: Simplify the remaining expression

Now we multiply the numbers together and apply the laws of exponents to the remaining variables: $$ \frac{8 \times 15 m n a^{7}}{3 \times 2 a^{5}} = \frac{120 m n a^{7}}{6 a^{5}} $$ We can simplify further by dividing the constant terms: $$ \frac{20 m n a^{7}}{a^{5}} $$ Since \(a^7\) and \(a^5\) have the same base, we can subtract their exponents to simplify further: $$ 20 m n a^{(7-5)} = 20 m n a^{2} $$

Step 5: Final Answer

The final answer after performing the division is: $$ 20 m n a^{2} $$

Key Concepts

Reciprocal

When dealing with algebraic division, an important concept is the reciprocal of a fraction. The reciprocal simply means flipping the numerator and the denominator. For example, the reciprocal of a fraction like \( \frac{2m}{15a^{7}b^{2}} \) is \( \frac{15a^{7}b^{2}}{2m} \). This becomes especially useful during division operations because instead of dividing by a fraction, you can multiply by its reciprocal. This method simplifies the process of division and is a fundamental step when performing fractional division in algebra.

Simplification

Simplification is all about making an expression easier to understand or solve. When we have a complex fraction like \( \frac{8 m^{2} n \times 15 a^{7} b^{2}}{3 a^{5} b^{2} \times 2 m} \), simplification allows us to cancel out common factors.

This involves:

• Identifying and removing terms that appear both in the numerator and denominator.
• Reducing numbers by division.

In the original problem, \( m \) and \( b^{2} \) are cancelled because they appear in both the numerator and denominator. This leaves us with a simpler expression which is quicker and easier to work with. This step is critical to streamline the solving process and ensure accuracy.

Laws of Exponents

The laws of exponents are rules that help simplify expressions involving powers of the same base. In this exercise, these laws are crucial for simplifying expressions like \( a^{7} \) and \( a^{5} \).

Key laws include:

• Multiplying with the same base: add the exponents (e.g., \( a^{m} \times a^{n} = a^{m+n} \)).
• Dividing with the same base: subtract the exponents (e.g., \( a^{m} / a^{n} = a^{m-n} \)).

In our problem, we apply these laws by simplifying \( a^{7} / a^{5} \) to \( a^{2} \). Understanding and applying the laws of exponents makes reducing expressions much simpler and is an essential tool in algebra.