Problem 63 - Intermediate Algebra

Question

Multiply. $$ x^{2 / 3}(x-2) $$

Step-by-Step Solution

Answer

The expression is multiplied to get \(x^{5/3} - 2x^{2/3}\).

Step 1: Distribute the Term

To multiply the expression, distribute the term outside the parenthesis, \(x^{2/3}\), to each term inside the parenthesis, \((x - 2)\). This means you need to multiply \(x^{2/3}\) by \(x\) and by \(-2\).

Step 2: Multiply Exponents of x

When multiplying like bases, add the exponents. The first term is \(x^{2/3} \cdot x^1\). Add the exponents: \(\frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3}\). So, the first term becomes \(x^{5/3}\).

Step 3: Multiply by Constant

For the second term, multiply \(x^{2/3}\) by \(-2\). Keep the base of the exponent the same and multiply the coefficient: \(-2 \cdot x^{2/3}\).

Step 4: Write the Final Expression

Combine both terms obtained from the multiplication to write the final expression: \(x^{5/3} - 2x^{2/3}\).

Key Concepts

Exponents

Exponents are an essential part of algebra. They allow you to express repeated multiplication compactly. For example, instead of writing \(x \cdot x\), you write \(x^2\). When dealing with exponents, it's crucial to understand a few basic rules.

• When you multiply terms that have the same base, you add the exponents together. For example, when you multiply \(x^{a}\) by \(x^{b}\), you simply add the exponents to get \(x^{a+b}\).
• Remember that \(x= x^1\). This is important for correctly applying the rules when you multiply expressions.
• Fractional exponents like \(x^{2/3}\) are another way of writing roots. Here, the numerator indicates the power, and the denominator indicates the root. So \(x^{2/3}\) means the cube root of \(x^2\).

When you multiply \(x^{\frac{2}{3}}\) and \(x\) in the exercise, you add the exponents \(\frac{2}{3}\) and \(1\) to obtain \(x^{\frac{5}{3}}\). This is a fundamental aspect of working with exponents.

Distributive Property

The distributive property is a useful tool in algebra. It allows you to multiply a single term across a sum or difference inside parentheses. This property is expressed as \(a(b + c) = ab + ac\).

• It's particularly handy when you need to simplify expressions without expanding into unnecessary complexity.
• Applying the distributive property means you will distribute the outside term to each term within the parentheses.

In the given problem, the task is to apply the distributive property to \(x^{\frac{2}{3}}(x - 2)\). You'll need to multiply \(x^{\frac{2}{3}}\) by each term inside the parenthesis, effectively simplifying and removing the parenthesis.When you distribute \(x^{\frac{2}{3}}\), you perform two multiplications:

• Multiply \(x^{\frac{2}{3}}\) by \(x\), which gives you \(x^{\frac{5}{3}}\).
• Multiply \(x^{\frac{2}{3}}\) by \(-2\), resulting in \(-2x^{\frac{2}{3}}\).

Simply put, the distributive property expands the expression, making it easier to solve or simplify further.

Multiplication of Expressions

When you multiply algebraic expressions, you’re combining several mathematical operations into one task. It's vital to understand how different aspects of expressions multiply together:

• Multiplying coefficients is straightforward: just multiply the numbers.
• When multiplying variables, apply the rules of exponents to combine similar terms correctly.

In this multiplication problem, you must first use the distributive property to identify the terms to be multiplied. The first multiplication is \(x^{\frac{2}{3}} \cdot x\), which involves adding the exponents to get \(x^{\frac{5}{3}}\). The second multiplication is \(-2 \cdot x^{\frac{2/3}}\), where the constant \(-2\) multiplies directly with \(x^{\frac{2/3}}\). Multiplying expressions might involve variables or numbers and require thoughtful application of algebraic rules to simplify or solve equations. But with practice, this becomes a quick method to solve complex problems.