Problem 41 - Algebra 2

Question

Write each expression in simplest form. Assume that all variables are positive. $$5\left(x^{\frac{2}{3}}\right)^{-1}$$

Step-by-Step Solution

Answer

The simplified form of the expression \( 5\left(x^{\frac{2}{3}}\right)^{-1}\) is \(5x^{-\frac{2}{3}}\)

Step 1: Identify the Negative Exponent

First, identify that \(x^{\frac{2}{3}}\) is raised to the -1 power. According to the rules of exponents, this means we have \(5 \div x^{\frac{2}{3}}\).

Step 2: Simplify the Division Operation

The division operation then can be written as a multiplication. This gives us \(5 * x^{-\frac{2}{3}}\).

Step 3: Simplify the Product

Lastly, we leave the expression in this form as this represents the simplest form in accordance with the rules of exponents.

Key Concepts

Negative Exponents

Negative exponents may seem tricky at first, but they follow a straightforward rule. Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, if you have a negative exponent like \( x^{-a} \), this can be rewritten as \( \frac{1}{x^a} \). Simply put, a negative exponent "flips" the base to the denominator if it's not already there. In the provided exercise, we saw \( x^{\frac{2}{3}} \) raised to the -1 power. By applying the rule of negative exponents, we transform this into \( \frac{1}{x^{\frac{2}{3}}} \). Instead of having \( x^{\frac{2}{3}} \) in the numerator, it now becomes part of the denominator.
Understanding this rule is key to simplifying expressions that involve negative exponents. It helps to rearrange and clarify the expression, making it easier to work with.

Rules of Exponents

The rules of exponents are a set of guidelines that help us simplify expressions involving powers of the same base. Let's look at some basic rules to clarify how these help us in simplification:
- **Product Rule:** \( x^a \cdot x^b = x^{a+b} \)
- **Quotient Rule:** \( \frac{x^a}{x^b} = x^{a-b} \)
- **Power of a Power Rule:** \((x^a)^b = x^{a \cdot b} \)
- **Negative Exponent Rule:** \( x^{-a} = \frac{1}{x^a} \)
- **Zero Exponent Rule:** \( x^0 = 1 \) if \( x eq 0 \)
Understanding these rules helps to rewrite expressions so they are easier to interpret and solve.
In our solution, the division in step 2 was rewritten as a multiplication using the negative exponent rule. The expression \( x^{-\frac{2}{3}} \) was easily simplified using these rules, showcasing their utility in simplifying complex fractional exponents.

Rational Exponents

Rational exponents are an extension of exponents to fractions. They represent roots and powers simultaneously. For example, \( x^{\frac{m}{n}} \) means the \( n \)-th root of \( x^m \). So, \( x^{\frac{2}{3}} \) in our expression signifies the cube root of \( x^2 \), or \( (\sqrt[3]{x})^2 \).
This concept is particularly useful in expressing roots in exponential form, allowing for easier manipulation, especially when combined with other exponent rules. When you encounter an expression like \( 5(x^{\frac{2}{3}})^{-1} \), understanding rational exponents lets you clearly see that \( x^{\frac{2}{3}} \) corresponds to both a root and a power.
This dual nature, when combined with other rules such as the negative exponent rule, enables straightforward simplification of more complex algebraic expressions. Rational exponents provide a powerful tool for expressing and manipulating different forms of roots and powers in algebra.