Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt[3]{x^{5}} $$

Step-by-Step Solution

Answer

The expression \( \sqrt[3]{x^{5}} \) can be rewritten as \( x^{5/3} \).

Step 1: Understand the Problem

We need to rewrite the expression \( \sqrt[3]{x^5} \) using rational exponents. A rational exponent is another way to represent roots and powers.

Step 2: Learn the Rational Exponent Rules

The expression \( \sqrt[n]{a^m} \) can be rewritten as \( a^{m/n} \). Here, the power \( m \) is the numerator and the root \( n \) is the denominator of the rational exponent.

Step 3: Apply the Rules to the Given Expression

We will apply the rule \( \sqrt[n]{a^m} = a^{m/n} \) to the given expression. By comparing, we see that \( n = 3 \) and \( m = 5 \). So, the expression \( \sqrt[3]{x^5} \) becomes \( x^{5/3} \).

Key Concepts

Cube Root

The cube root of a number is finding a value that, when multiplied by itself three times, gives the original number. It's like asking what number cubed equals our original number. For example, the cube root of 8 is 2 because

• 2 × 2 × 2 = 8.

The cube root can also be expressed using a radical symbol, such as \( \sqrt[3]{x} \). This is specifically for cube roots.
While we often deal with square roots, cube roots appear when working with three-dimensional quantities or in higher-level algebra problems. Using rational exponents offers a more flexible way to handle cube roots, allowing for easier algebraic manipulations.

Exponentiation

Exponentiation is a way of expressing repeated multiplication of the same number. When we write \( x^5 \), we're saying multiply \( x \) by itself five times. It’s a shorthand that simplifies expressions and calculations.
Here’s a quick recap of how exponentiation works:

• Base: The number being multiplied (\( x \) in \( x^5 \)).
• Exponent: The number of times the base is multiplied by itself (5 in \( x^5 \)).

Exponents can be whole numbers, which represent repeated multiplication. They can also be fractions, which represent roots, like in the expression \( x^{5/3} \). This allows us to express roots and powers in the same kind of notation.

Algebra

Algebra is a foundational branch of mathematics that deals with symbols and the rules for manipulating these symbols. In the context of expressions like \( \sqrt[3]{x^5} \), algebra helps us to write and solve equations in simpler ways.
This involves:

• Understanding expressions and transforming them using algebraic rules, such as the rules for rational exponents.
• Simplifying expressions to make calculations more straightforward.
• Solving equations that involve unknowns, like \( x \) in \( x^{5/3} \).

Algebra allows for the generalization of arithmetic operations and helps in solving problems through abstraction, making it easier to handle complex equations.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. In mathematical notation, they are typically written as \( \frac{a}{b} \), where

• \( a \) and \( b \) are integers.
• \( b eq 0 \) since division by zero is undefined.

Rational exponents, like in \( x^{5/3} \), fit into this concept because the exponent \( \frac{5}{3} \) is a rational number.
This notation helps us understand and perform operations involving roots and powers systematically, allowing us to simplify complex expressions and solve equations more easily.