Problem 94 - Elementary Algebra

Question

Problem: Simplify: \(\left(7 n^{4}\right)\left(9 n^{6}\right)\) $$ \text { Incorrect Answer: } \begin{aligned} &\left(7 n^{4}\right)\left(9 n^{6}\right) \\ &=63 n^{(4)(6)} \\ &=63 n^{24} \end{aligned} $$ 95\. 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

1st problem: 63 n^{10}; 2nd problem: 5 n^{8}

Step 1: Identify correcting errors in first problem

Observe the initial answer is: \[\left(7 n^{4}\right)\left(9 n^{6}\right) = 63 n^{(4)(6)} = 63 n^{24}\]This is incorrect because when multiplying exponents with the same base, the exponents are added, not multiplied.

Step 2: Recompute exponents in first problem correctly

Correct the exponents combination: \[\left(7 n^{4}\right)\left(9 n^{6}\right) = 7\cdot9 \cdot n^{4+6} = 63 n^{10}\]

Step 3: Identify correcting errors in second problem

Observe the initial answer is: \[\frac{15 n^{12}}{3 n^{4}} = 5 n^{12+4} = 5 n^{3}\]This is incorrect because when dividing exponents with the same base, the exponents are subtracted, not added.

Step 4: Recompute exponents in second problem correctly

Correct the exponents combination: \[\frac{15 n^{12}}{3 n^{4}} = \frac{15}{3} \cdot \frac{n^{12}}{n^{4}} = 5\cdot n^{12-4} = 5 n^{8}\]

Key Concepts

Exponent Rules

Understanding the fundamental rules of exponents is crucial for simplifying expressions. Exponents are shorthand for repeated multiplication of the same number by itself. For example, in the expression \(a^3\), the base \(a\) is multiplied by itself three times: \(a \times a \times a\). The core rules of exponents include:

• Multiplication Rule: When multiplying like bases, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Division Rule: When dividing like bases, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power Rule: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\).
• Zero Exponent Rule: Any base raised to the exponent zero is equal to one: \(a^0 = 1\) for \(a eq 0\).

Mastering these rules helps in correctly simplifying complex algebraic expressions involving exponents.

Multiplication of Exponents

When you multiply terms with the same base, you add their exponents. Let's look at an example. Given the expression \( (7n^4)(9n^6) \), follow these steps:

• First, multiply the numeric coefficients (constants): \(7 \times 9 = 63\).
• Second, add the exponents of \(n\): \( n^{4+6} = n^{10} \).

So the correct simplified expression is \(63n^{10}\). Remember, the key is to add the exponents of the same base.

Division of Exponents

Dividing expressions with exponents involves subtracting the exponents of like bases. For instance, look at the following expression: \( \frac{15n^{12}}{3n^{4}} \). The steps to simplify it are:

• First, divide the constants: \( \frac{15}{3} = 5 \).
• Next, subtract the exponents of \(n\): \( n^{12-4} = n^{8} \).

So, the simplified form is \( 5n^8 \). The key rule here is to subtract the exponents when dividing terms with the same base.

Simplifying Expressions

Simplifying algebraic expressions with exponents involves correctly applying exponent rules. Let's recap with our examples:

• For multiplication: Given \( (7n^4)(9n^6) \), we multiply the constants and add the exponents to get \( 63n^{10} \).
• For division: From \( \frac{15n^{12}}{3n^{4}} \), we divide the constants and subtract the exponents to obtain \( 5n^8 \).

Make sure to:

• Identify like bases before applying the rules.
• Accurately perform the operations with exponents (addition or subtraction).

With these steps, simplifying becomes straightforward and manageable, ensuring correct results.