Problem 40 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The simplest form of the expression \((3x^{2/3})^{-1}\) is \[1 / (3x^{2/3})\]

Step 1: Identify exponent properties

Firstly, recognize that the outer exponent of \(-1\) implies taking the reciprocal of the base, \(3x^{2/3}\). Additionally, remember that if an exponent (like \(2/3\) here) is raised to another power (like \(-1\) here), the exponents get multiplied.

Step 2: Apply the negative exponent rule

Apply the rule that any nonzero number raised to the power of \(-1\) is its reciprocal. In this case, \((3x^{2/3})^{-1}\) is the reciprocal of \(3x^{2/3}\). The reciprocal of a number is 1 divided by that number. So, this gives us \[1 / (3x^{2/3})\].

Step 3: Simplify the expression

To finalise the simplification, keep the coefficient (3) in the denominator and apply the exponent \(2/3\) to the variable \(x\), leading to the final simplified form of our expression: \[1 / (3x^{2/3})\]

Key Concepts

Exponent Properties

Exponents are a crucial concept in algebra that dictate how many times a number, known as the base, is multiplied by itself. There are a few key properties of exponents that help simplify expressions:

• Multiplication of Exponents: When you multiply expressions with the same base, you add their exponents. For example, \( a^m \times a^n = a^{m+n} \).
• Power of a Power: When you raise a power to another power, you multiply the exponents. Therefore, \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product: When a product is raised to an exponent, you distribute the exponent to each factor in the product. Thus, \( (ab)^n = a^n b^n \).

Understanding these properties is crucial for simplifying algebraic expressions involving exponents. In our exercise, we applied the power of a power rule to simplify the exponent \((2/3)\) raised to \(-1\), resulting in \(x^{-2/3}\). This way, operations become more manageable, making it easier to reach the simplest form.

Negative Exponent Rule

The negative exponent rule is another vital tool that serves to simplify expressions involving exponents. This rule states that for any non-zero base \(a\), \(a^{-n} = 1/a^n\). In simple terms, a negative exponent signifies the reciprocal of the base raised to the opposite (positive) exponent.

In the given exercise, this rule was used to transform the expression \((3x^{2/3})^{-1}\) into its reciprocal, giving us \(1/(3x^{2/3})\).

• By applying the negative exponent rule, you flip the base, making it easier to simplify further.
• Remember, this rule is valid for all positive bases and all real numbers \(n\).

This transformation simplifies the expression while still preserving its original value, necessary for critical higher-level algebraic manipulations.

Reciprocal

In mathematics, a reciprocal is intuitively the 'flipping' of a fraction. If you have a number written as a fraction, such as \(a/b\), its reciprocal is \(b/a\). This concept helps in dealing with expressions involving negative exponents.

The reciprocal has some important characteristics:

• Multiplicative Inverse: A number multiplied by its reciprocal equals one. For example, \(a \times (1/a) = 1\).
• Calculation: To find the reciprocal of a fraction or expression, switch the numerator and the denominator's positions.

In our problem, understanding the reciprocal allowed us to simplify \((3x^{2/3})^{-1}\) by taking it as \(1/(3x^{2/3})\). Thus applying this concept helped us represent the expression clearly, making calculations more straightforward.