Problem 43 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ 21 m^{4} n^{2} \div \frac{3 m n^{2}}{7 n} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(21 m^{4} n^{2} \div \frac{3 m n^{2}}{7 n}\) Answer: \(49 m^{3} n\)

Step 1: Rewrite the expression as a multiplication problem

Rewrite the expression by inverting the fraction and changing the division sign to a multiplication sign: $$ 21 m^{4} n^{2} \div \frac{3 m n^{2}}{7 n} = 21 m^{4} n^{2} \times \frac{7 n}{3 m n^{2}} $$

Step 2: Combine the coefficients and the variables

Multiply the coefficients (the numbers) and the variables (the exponents) separately: $$ (21 \times \frac{7}{3}) (m^{4} \times \frac{1}{m})(n^{2} \times \frac{n}{n^{2}}) $$

Step 3: Simplify the coefficients

Multiply the coefficients: $$ \frac{21 \times 7}{3} = \frac{147}{3} = 49 $$

Step 4: Simplify the variables

Apply the rules of exponents when multiplying variables with the same base: For m: $$ m^{4} \times \frac{1}{m} = m^{4-1} = m^3 $$ For n: $$ n^{2} \times \frac{n}{n^{2}} = n^{2-2} \times n^{1} = n^{0} \times n = 1 \times n = n $$

Step 5: Combine the results

Combine the simplified coefficients and variables to find the final answer: $$ 49 m^{3} n $$ Thus, the simplified expression is \(49 m^{3} n\).

Key Concepts

Multiplying Fractions

When multiplying fractions, it is important to understand that each component of the fractions has a role to play.

• The numerator is the top part of the fraction, and the denominator is the bottom part.
• To multiply fractions, simply multiply the numerators together to get a new numerator.
• Do the same for the denominators to get a new denominator.

For example, when dealing with the equation \( 21 \times \frac{7}{3} \), we treat 21 as a fraction by imagining its denominator as 1: \( \frac{21}{1} \times \frac{7}{3} \).This yields \( \frac{21 \times 7}{1 \times 3} = \frac{147}{3} \), which can be simplified to 49.
By simplifying, we ensure fractions don't stay unnecessarily complex.

Dividing Fractions

Dividing fractions might initially seem complicated, but it becomes quite straightforward once you learn the key rule: the division of fractions is simply the multiplication by the reciprocal of the divisor.

• The reciprocal of a fraction is created by swapping its numerator and denominator.
• Thus, dividing by a fraction is the same as multiplying by its reciprocal.

For example, in the original exercise, \( 21 m^{4} n^{2} \div \frac{3 m n^{2}}{7 n} \) becomes \( 21 m^{4} n^{2} \times \frac{7 n}{3 m n^{2}} \).By flipping the divisor and changing to multiplication, you can use multiplication rules to find the results.

Exponent Rules

In algebra, handling exponents is key to simplifying expressions accurately. These rules make dealing with powers straightforward and logical.

• When multiplying terms with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
• When dividing terms with the same base, subtract the exponents: \( a^m \div a^n = a^{m-n} \).
• Remember, any base raised to the power of zero is one, provided the base is not zero, as in \( a^0 = 1 \).

In the exercise, for the expression \( m^4 \times \frac{1}{m} \), we apply \( m^{4-1} \) to get \( m^3 \).For \( n^2 \times \frac{n}{n^2} \), since \( n^2-2+1 = 1 \), it simplifies to \( n \).Using these rules helps manage complex equations effectively.