Problem 67 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \frac{\left(x^{n+2}\right)^{3}}{x^{2 n}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((x^{n+2})^3 / x^{2n}\) using the power rules for exponents. Answer: The simplified expression is \(x^{n+6}\).

Step 1: Apply the power rule for \((a^m)^n = a^{m*n}\)

Rewrite the expression as \((x^{(n+2)*3}) / x^{2n}\), which simplifies to \((x^{3n+6}) / x^{2n}\).

Step 2: Apply the exponent rule for \(a^{m}/a^{n}=a^{m-n}\)

Since the bases are the same, we can subtract the exponent in the denominator from the exponent in the numerator: \(x^{(3n+6)-2n}\).

Step 3: Simplify the exponent

To find the simplified exponent, subtract 2n from the exponent in the numerator: (3n + 6) - 2n. This simplifies to n+6.

Step 4: Write the final simplified expression

Plug the simplified exponent back into the expression: \(x^{n+6}\). The simplified expression is: $$ x^{n+6} $$

Key Concepts

Power Rule

The power rule is an essential tool for simplifying expressions with exponents. When you have a base raised to a power and that whole expression is raised to another power, the power rule comes into play. It can be expressed as \((a^m)^n = a^{m \cdot n}\). Basically, you multiply the exponents together. This method helps make complex expressions simpler and more manageable. For instance, in the original problem, we have \((x^{n+2})^3\). By using the power rule, we can transform this into \(x^{(n+2) \cdot 3}\). After simplifying, it becomes \(x^{3n+6}\). This step is crucial, as it helps break down larger exponent expressions into easier forms. Power rule mastery is a key skill for anyone learning algebraic manipulations.

Exponentiation

Exponentiation refers to the operation of raising a number, or base, to the power of an exponent. This operation is a shorthand way to express repeated multiplication, such as \(2^3\), which means \(2 \times 2 \times 2\). It's a basic concept in mathematics that helps streamline complex calculations. In algebra, exponentiation with variables follows the same principles. If you seen an expression like \(x^{n+2}\), it signifies that the base \(x\) is multiplied by itself \(n+2\) times.

Understanding these fundamentals allows you to tackle more intricate algebraic concepts, like expressions containing multiple exponential terms. It lays the groundwork for applying rules, such as the power rule or the division of exponents. Always ensure your bases are consistent, and handle the exponents according to the appropriate rules to simplify the expressions accurately.

Algebraic Simplification

Algebraic simplification involves using mathematical rules to rewrite expressions in a more concise form. The goal is to make the expression easier to understand or solve. In dealing with expressions like \(\frac{(x^{3n+6})}{x^{2n}}\), algebraic simplification means applying the rules of exponents to combine and reduce them.

The key rule in this context is \(a^m / a^n = a^{m-n}\), which is used to simplify divisions involving the same bases. It allows us to subtract exponents from each other when we divide terms. Applying this to our problem, after using the power rule, we get \(x^{(3n+6) - 2n}\), which simplifies to \(x^{n+6}\). By reducing the exponents through subtraction, we decrease the complexity of the expression without changing its value. This practice is fundamental not just in algebra, but in all areas of mathematics. It aids in solving equations more efficiently and understanding mathematical relationships more clearly.