Problem 86 - Algebra 2

Question

Simplify. Assume that no variable equals zero. \(\frac{a^{4} n^{7}}{a^{3} n}\)

Step-by-Step Solution

Answer

The simplified expression is \( a n^6 \).

Step 1: Identify the Expression

The expression given to us is \( \frac{a^{4} n^{7}}{a^{3} n} \). Our goal is to simplify this expression by applying the properties of exponents.

Step 2: Simplify the Exponent of 'a'

In the expression, focus on the powers of 'a'. We have \( a^4 \) in the numerator and \( a^3 \) in the denominator. According to the rule \( \frac{a^m}{a^n} = a^{m-n} \), subtract the exponent in the denominator from the exponent in the numerator: \( 4 - 3 = 1 \). Thus, \( a^4 / a^3 = a^1 = a \).

Step 3: Simplify the Exponent of 'n'

Now, let's focus on the powers of 'n'. We have \( n^7 \) in the numerator and \( n^1 \) (or just \( n \)) in the denominator. Using the same exponent rule, subtract the exponent in the denominator from the exponent in the numerator: \( 7 - 1 = 6 \). Thus, \( n^7 / n = n^6 \).

Step 4: Combine the Simplified Components

Combine the results from Steps 2 and 3: \( a \) and \( n^6 \). The simplified form of the original expression is \( a n^6 \).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when simplifying algebraic expressions. Exponents, also known as powers, are used to express repeated multiplication of a number by itself. For example, \( a^3 \) represents \( a \times a \times a \). Exponential properties provide a set of guidelines that enable us to manipulate and simplify expressions in a systematic way. Some key properties include:

• Product of Powers Property: When multiplying with the same base, you add the exponents. For instance, \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Property: When dividing with the same base, you subtract the exponents. Therefore, \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Property: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).

By applying these properties to expressions like the one in our problem, we can simplify complex fractions and create more manageable terms.

Fractional Exponents

Fractional exponents can seem daunting at first, but they follow the same basic principles as integer exponents. A fractional exponent such as \( a^{1/2} \) is another way of expressing roots. Specifically, \( a^{1/2} \) is equivalent to the square root of \( a \), and \( a^{1/3} \) represents the cube root. In general, the expression \( a^{m/n} \) is equivalent to \( \sqrt[n]{a^m} \). This means the denominator of the fraction indicates the root, while the numerator indicates the power.Understanding fractional exponents is essential for simplifying expressions. When expressions contain variables with fractional exponents, the same rules apply:

• Multiplying with Fractional Exponents: Add the exponents as you would with integers, following \( a^{m/n} \times a^{p/q} = a^{\frac{mq + np}{nq}} \).
• Dividing with Fractional Exponents: Similar to integer exponents, subtract them: \( \frac{a^{m/n}}{a^{p/q}} = a^{\frac{mq - np}{nq}} \).

Grasping these subtleties will help in understanding more advanced algebraic manipulations.

Exponent Rules

Exponent rules are vital for making sense of algebraic operations, especially when simplifying expressions. Let's quickly revisit the critical exponent rules used in our original problem:**Simplifying "a" and "n":**- Using the quotient rule, for the expression \( \frac{a^4}{a^3} \), you subtract the exponents. This gives us \( a^{4-3} = a^1 = a \).- Similarly, for \( \frac{n^7}{n} \) or \( \frac{n^7}{n^1} \), subtract the exponents to get \( n^{7-1} = n^6 \).These steps show how the power of exponents simplifies the division of similar base terms. Beyond these examples, here are other crucial exponent rules to remember:

• Zero Exponent Rule: Any base raised to the zero power is one: \( a^0 = 1 \), provided \( a eq 0 \).
• Negative Exponent Rule: An exponent with a negative sign signifies taking the reciprocal of the base raised to the positive exponent: \( a^{-m} = \frac{1}{a^m} \).

Mastering these exponent rules can greatly aid in solving a variety of algebraic problems efficiently and accurately.