Problem 67 - Algebra 2

Question

Simplify each expression. Assume that all variables are positive. $$x^{\frac{3}{5}} \div x^{\frac{1}{10}}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{\frac{1}{2}}\). If you choose to simplify further, note that \(x^{\frac{1}{2}}\) is the same as \(\sqrt{x}\).

Step 1: Identify the Problem

The task is to simplify the expression \(x^{\frac{3}{5}} \div x^{\frac{1}{10}}\). The law of exponents that applies is the quotient rule which states that \(a^{m} \div a^{n} = a^{m-n}\) where m and n are the exponents and a is the common base.

Step 2: Apply the Quotient Rule

Using the quotient rule, subtract the exponent \(\frac{1}{10}\) from \(\frac{3}{5}\) which gives \(x^{\frac{3}{5} - \frac{1}{10}}\). Note that the base x remains unchanged.

Step 3: Perform the Subtraction

Subtract the fractions \(\frac{3}{5} - \frac{1}{10}\) by finding a common denominator. The least common multiple (LCM) of 5 and 10 is 10. Write both fractions with a denominator of 10 and subtract: \(\frac{6}{10} - \frac{1}{10} = \frac{5}{10}\). Thus the expression simplifies to \(x^{\frac{5}{10}}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an important skill in mathematics, as it allows us to write complex expressions in a simpler form. This often involves reducing the expression to its most basic components.
In our example, we have the expression \(x^{\frac{3}{5}} \div x^{\frac{1}{10}}\). The goal is to combine the powers of \(x\) into a single exponent, making the expression easier to interpret or solve.
When simplifying expressions like this, one useful tool is the law of exponents, which gives us rules to handle powers and roots efficiently. In the context of division, we use the quotient rule of exponents, which allows us to simplify expressions involving powers of the same base.

Fraction Subtraction

Subtracting fractions is a key skill needed to apply the quotient rule of exponents, as fractional exponents are common in algebra.
When subtracting fractions, you must find a common denominator. This is crucial because it standardizes the fractions, enabling subtraction.
In our exercise, we dealt with \(\frac{3}{5}\) and \(\frac{1}{10}\). The least common multiple of these denominators is 10.
Convert \(\frac{3}{5}\) to \(\frac{6}{10}\) by multiplying both the numerator and denominator by 2. Then you can subtract the fractions:

• \(\frac{6}{10} - \frac{1}{10}\)

Simply subtract the numerators while keeping the denominator the same. This results in \(\frac{5}{10}\), or simplified further, \(\frac{1}{2}\).
Understanding this fundamental process enables the simplification of the original expression according to the quotient rule.

Law of Exponents

The law of exponents provides a set of rules for manipulating expressions with exponents, which are crucial for algebra and higher-level math. One of the key rules is the quotient rule, which we used in our exercise.
The quotient rule states that \(a^{m} \div a^{n} = a^{m-n}\), where \(a\) represents a common base and \(m\) and \(n\) are the exponents.
In our example, applying this rule to \(x^{\frac{3}{5}} \div x^{\frac{1}{10}}\) means we subtract the exponents to get \(x^{\frac{3}{5} - \frac{1}{10}}\), which simplifies to \(x^{\frac{1}{2}}\) after performing the fraction subtraction.
This rule simplifies division problems involving exponential expressions by reducing them to a single power, making calculations more straightforward.
Mastering the laws of exponents like the quotient rule is essential, as it opens the door to simplifying and solving a wide range of algebraic expressions.