Problem 101 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$\frac{3^{7 / 6}}{3^{1 / 6}}$$

Step-by-Step Solution

Answer

The simplified expression is \(3^1\), which is simply 3.

Step 1: Identify the rule

There is a rule in exponents which states that \(a^m / a^n = a^{m-n}\). This rule applies when you divide terms with the same base. In this case, 3 is the common base.

Step 2: Apply the rule

Applying the subtraction rule of exponents:\n\[\frac{3^{7 / 6}}{3^{1 / 6}} = 3^{(7 / 6)-(1 / 6)}\]

Step 3: Simplify the expression

Subtracting the exponents:\n\[3^{(7 / 6)-(1 / 6)} = 3^{6 / 6} = 3^1\]

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that involves reducing expressions to their simplest form. This helps us deal with fewer terms and less complicated mathematics. When we simplify, we aim to express a complex expression in a way that is easier to understand and use.

In our example, we have the expression \( \frac{3^{7/6}}{3^{1/6}} \). Simplification often involves using different rules of exponents, like combining like terms or dealing with division and multiplication of powers. By following the procedures in rules of exponents, we make the expression simpler and more straightforward to evaluate.

Division of Exponents

When it comes to dealing with exponents, the division rule is another handy tool. This rule tells us how to divide powers with the same base, which is crucial in reducing expressions efficiently. If we have a fraction such as \( \frac{a^m}{a^n} \), it's equivalent to \( a^{m-n} \). This rule only applies when the bases are the same.

For our specific problem, the bases of both terms, \( 3^{7/6} \) and \( 3^{1/6} \), are both 3. This allows us to use the division rule to simplify the expression to a lower power of the base, making further calculations much simpler.

Subtracting Exponents

Subtracting exponents is a key process when using the division rule of exponents. It helps us condense expressions with multiple powers into a single power. By subtracting the exponents, we are effectively computing the power difference within the division.

In the given exercise, after identifying the common base, we subtract the exponents: \( 3^{(7 / 6)-(1 / 6)} \). This gives us the exponent \( 6/6 \), and when simplified, it becomes \( 3^1 \). Subtracting exponents simplifies the expression considerably, reducing complexity and making it easier to manage.

Common Base in Exponents

Using a common base is central to applying the rules of division and subtraction of exponents. When the base is the same in both the numerator and the denominator, it allows us to simplify the expression easily using these rules.

For this exercise, the base 3 is common in both parts of the expression \( \frac{3^{7/6}}{3^{1/6}} \). This commonality enables the straightforward application of exponent rules, ultimately simplifying the full expression to \( 3^1 \). Recognizing a common base is crucial since it determines whether the division rule and other exponent rules can be appropriately applied.