Problem 73 - Algebra 2

Question

Simplify each expression. Assume that all variables are positive. $$\left(x^{\frac{1}{2}} \cdot x^{\frac{5}{12}}\right)^{\frac{1}{3}} \div x^{\frac{2}{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{-\frac{4}{9}}\).

Step 1: Combine exponents of multiplication

Combine the exponents in the parentheses using the rule of addition. This gives \(x^{\frac{1}{2} + \frac{5}{12}} = x^{\frac{8}{12}} = x^{\frac{2}{3}}\). Thus the expression becomes \((x^{\frac{2}{3}})^{\frac{1}{3}} \div x^{\frac{2}{3}}\).

Step 2: Simplify the exponential of a power

This should be simplified by multiplying the exponents using the rule \((x^a)^b = x^{a \cdot b}\). This gives \((x^{\frac{2}{3}})^{\frac{1}{3}} = x^{\frac{2}{3} \cdot \frac{1}{3}} = x^{\frac{2}{9}}.\). Thus the expression becomes \(x^{\frac{2}{9}} \div x^{\frac{2}{3}}\).

Step 3: Simplify the division of exponents of the same base

Use the rule of subtraction to simplify \(x^{\frac{2}{9}} \div x^{\frac{2}{3}} = x^{\frac{2}{9} - \frac{2}{3}}\). This gives \(= x^{\frac{2}{9} - \frac{6}{9}} = x^{-\frac{4}{9}}\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves finding a way to make a complex expression easier to work with, often by reducing the number of terms or simplifying the powers. Imagine having a complex expression that looks like it’s needing to be tamed. In our exercise, we begin with a mix of fractional exponents. The first step is to take care of the multiplication of like bases. This process allows us to simplify the expression significantly.

For example, if you have \(x^{\frac{1}{2}} \cdot x^{\frac{5}{12}}\), you combine the exponents by adding them, since both terms have the same base. The result: \(x^{\frac{1}{2} + \frac{5}{12}}\) simplifies to \(x^{\frac{8}{12}}\), which further reduces to \(x^{\frac{2}{3}}\). This step simplifies our original expression into a more manageable form.

Simplifying expressions with exponents helps make calculations easier and clearer, keeping the expression neat and organized.

Properties of Exponents

Understanding the properties of exponents is crucial when simplifying expressions with powers. These properties act like tools in your math toolbox. They help you manipulate expressions to your advantage.

Some general rules include:

• Adding exponents when multiplying like bases: \(x^a \cdot x^b = x^{a+b}\)
• Multiplying exponents when raising a power to a power: \((x^a)^b = x^{a\cdot b}\)
• Subtracting exponents when dividing like bases: \(x^a \div x^b = x^{a-b}\)

The above rules allow us to turn complicated expressions into simple ones. In our example, once the multiplication in the parenthesis is simplified, we apply the power of a power rule: \((x^{\frac{2}{3}})^{\frac{1}{3}}\). This step requires multiplying the exponents, and transforms the expression to \(x^{\frac{2}{9}}\).

These properties are vital for obtaining an expression that is as simplified as possible, making further calculations easier!

Fractional Exponents

Fractional exponents may seem unfamiliar at first, but they're just a compact way to express roots and powers. Instead of writing \((\sqrt{x})\), we use exponents. For instance, \(x^{\frac{1}{2}}\) represents the square root of \(x\), while \(x^{\frac{1}{3}}\) indicates the cube root.

When combining fractional exponents, the arithmetic is the same as with whole numbers. You add them when multiplying or subtract when dividing. In our example, the division \(x^{\frac{2}{9}} \div x^{\frac{2}{3}}\) simplifies by subtracting the exponents. This results in \(x^{\frac{2}{9} - \frac{6}{9}} = x^{-\frac{4}{9}}\). Here, \(x^{-\frac{4}{9}}\) signals that we have an inverse, or reciprocal, involved.

By understanding fractional exponents, you gain a deeper grasp of how roots and powers relate, allowing for elegant solutions to complex problems.