Problem 54 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[7]{(3 x-y)^{4}}\)

Step-by-Step Solution

Answer

\((3x-y)^{\frac{4}{7}}\).

Step 1: Understand the Root and Its Exponent Form

The expression given is \( \sqrt[7]{(3x-y)^4} \). The root is a 7th root, which means it can be converted to an exponent as \( (3x-y)^4 \) raised to the power of \( \frac{1}{7} \). This is because the nth root of a number is equivalent to raising that number to the power of \( \frac{1}{n} \).

Step 2: Apply the Laws of Exponents

According to the laws of exponents, if you have an expression \( (a^m)^n \), it can be simplified to \( a^{m \times n} \). In this exercise, we first have \( (3x-y)^4 \) raised to the power of \( \frac{1}{7} \). By the laws of exponents, this composition becomes \( (3x-y)^{4 \times \frac{1}{7}} \).

Step 3: Calculate the New Exponent

Multiply the inner exponent 4 by the outer exponent \( \frac{1}{7} \). The calculation is straightforward: \( 4 \times \frac{1}{7} = \frac{4}{7} \). Thus, the entire expression becomes \( (3x-y)^{\frac{4}{7}} \).

Key Concepts

Roots and Radicals

Roots and radicals are fundamental concepts in algebra, providing a different way to represent recurring mathematical operations. They often show up in equations where numbers or algebraic expressions are divided into equal parts. When you see a root sign, you are looking at a radical. Here’s how it works:

For the expression \( \sqrt[7]{a} \), this is known as the 7th root of \( a \). It means you're trying to find a value that, when multiplied by itself seven times, equals \( a \). The root symbol \( \sqrt{} \) helps express these operations, but sometimes it's easier to work with exponents instead of radicals. That’s where rational exponents come in handy.

The transition from a root to an exponent follows this idea: the nth root of \( a \) is the same as \( a^{\frac{1}{n}} \). So, \( \sqrt[7]{(3x-y)^4} \) becomes \( (3x-y)^4 \) raised to the power of \( \frac{1}{7} \). When you get comfortable with turning roots into exponents, it opens up many simplification and solving opportunities.

Laws of Exponents

The laws of exponents are sets of rules that help you simplify expressions involving powers. Knowing these can make working with even the most complex expressions manageable. When dealing with exponents, you often encounter situations like combining exponents, raising powers to powers, and distributing over multiplication or division.

The most useful law for this exercise states that when you have an exponent raised to another exponent, you multiply the exponents. This is written as \( (a^m)^n = a^{m \times n} \).

In our problem, \( (3x-y)^4 \) gets raised to the power \( \frac{1}{7} \). The law tells us to multiply 4 by \( \frac{1}{7} \), resulting in \( (3x-y)^{\frac{4}{7}} \). Understanding and using these laws allow you to transform expressions into simpler or more useful forms, crucial for solving equations or exploring higher-level math.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves rewriting equations or expressions to make them easier to understand or solve. When simplifying, you aim to express the equation in its simplest form without changing its value.

For expressions with rational exponents, like \( (3x-y)^{\frac{4}{7}} \), simplifying involves several steps:

• Convert roots into rational exponents, making them easier to handle using the laws of exponents.
• Apply the laws of exponents to combine or reduce powers.
• Simplify the coefficients and variables when possible.

By following these steps, you transform a complex expression into its essence, like going from \( \sqrt[7]{(3x-y)^4} \) to \( (3x-y)^{\frac{4}{7}} \). Simplification not only makes calculations more straightforward but also reveals the underlying relationships between numbers and variables, a fundamental part of mastering algebra.