Problem 95 - Elementary Algebra

Question

Problem: Simplify: \(\frac{15 n^{12}}{3 n^{4}}\) $$ \text { Incorrect Answer: } \begin{aligned} & \frac{15 n^{12}}{3 n^{4}} \\ &=5 n^{12+4} \\ &=5 n^{3} \end{aligned} $$

Step-by-Step Solution

Answer

The correct simplification is \( 5 n^{8} \).

Step 1 - Identify the incorrect elements

Notice the problem in the incorrect answer. The exponent of the denominator should be subtracted from the exponent of the numerator, not added.

Step 2 - Divide the coefficients

Divide the coefficients to simplify the fractional part: \(\frac{15}{3} = 5\).

Step 3 - Simplify the exponents

Subtract the exponent in the denominator from the exponent in the numerator for the variable \( n \): \( n^{12-4} = n^{8} \).

Step 4 - Write the final expression

Combine the simplified coefficient and variable expression: \( 5 n^{8} \).

Key Concepts

Exponent Rules

Exponents represent repeated multiplication of the same number. For example, when you see something like \( n^{12} \), it means \( n \) is being multiplied by itself 12 times. In algebra, there are several rules to handle exponents effectively. One key rule relevant to the problem is the rule for dividing exponents: when you divide exponential expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator. For instance, in \( \frac{n^{12}}{n^{4}} \), you subtract 4 from 12, resulting in \( n^{8} \). This is because:

• \( n^{12} = n \times n \times \text{(12 times)} \)
• \( n^{4} = n \times n \times \text{(4 times)} \)

When divided, 4 of the 'n's in the dominator cancel out 4 of the 'n's in the numerator, leaving 8 'n's multiplied together. Remember to always apply this principle when simplifying expressions with exponents.

Coefficient Division

Coefficients are the numerical parts of terms that include variables. In the problem at hand, the coefficients are 15 and 3. To simplify the expression correctly, you need to divide these coefficients separately from the variable part:

• Divide the coefficients just as you would with regular numbers.
• In our example, divide 15 by 3, which results in 5.

Combining this with the previous step on exponents, you handle the numbers and variables separately first and then combine your results into a single simplified expression. This method ensures clarity and accuracy in your solution.

Variable Expressions

Variable expressions are combinations of numbers, variables (like \( n \)), and arithmetic operations. Properly understanding and simplifying these expressions requires a clear understanding of both the numbers and the variables involved. In our example, the variable part \( n \) and its associated exponents need special attention. Here’s a process to tackle such problems:

• Simplify the coefficients: 15 divided by 3 yields 5.
• Simplify the exponents using exponent rules: \( n^{12} \) divided by \( n^{4} \) gives \( n^{8} \).

Therefore, you combine the simplified coefficient and the simplified variable part to get the final expression: \( 5n^{8} \). Always write the coefficients first, followed by the variable with its new exponent. This helps keep your work organized and correct.