Problem 16 - Algebra 2

Question

Write each expression in radical form. $$x^{1.5}$$

Step-by-Step Solution

Answer

\(\sqrt{x}^3\)

Step 1: Converting the Decimal Exponent to a Fraction

The decimal exponent is \(1.5\), which can be converted into a fraction. In terms of fractions, \(1.5 = 3/2\). Therefore, we can rewrite \(x^{1.5}\) as \(x^{3/2}\).

Step 2: Converting to Radical Form

With the exponent as a fraction, we can easily convert the expression to a radical form. An expression of the form \(x^{n/m}\) can be written as \(\sqrt[m]{x^n}\) or \(\sqrt[m]{x}^n\). Applying this rule to \(x^{3/2}\), we get \(\sqrt[2]{x^3}\) or \(\sqrt{x}^3\). It is more conventional to omit the \(2\) in \(\sqrt[2]{x}\) and just write \(\sqrt{x}\). Therefore, the final answer in radical form is \(\sqrt{x}^3\).

Key Concepts

Understanding Decimal Exponents

Decimal exponents are exponents that are not whole numbers, and include numbers like 0.5, 1.2, or 3.7. These exponents are important because they represent more complex roots and powers than those defined by integer exponents. For example, an exponent of 1.5 indicates one and a half times the multiplication of a base. Decimal exponents often result in fractional power expressions that can be converted to radicals, providing a tangible representation of the exponentiation.

To work effectively with decimal exponents, it’s crucial to be comfortable converting decimals into fractions. This conversion allows for easier manipulation and understanding, particularly when translating into radical expressions.

Exploring Fractional Exponents

Fractional exponents represent the roots of numbers and are expressed in the form of fractions, such as \(a^{m/n}\), where \(m\) and \(n\) are integers. The numerator \(m\) signifies the power to which the number is raised, and the denominator \(n\) represents the root.

When dealing with fractional exponents:

• The expression \(x^{1/2}\) is equivalent to the square root of \(x\), \(\sqrt{x}\).
• \(x^{2/3}\) implies the cubic root of \(x^2\), expressed as \(\sqrt[3]{x^2}\).

Knowing how to work with fractional exponents is crucial for converting expressions into radicals and is widely used in various areas of mathematics, allowing for simplification and solving of complex problems.

Converting Exponents to Radicals

Converting exponents to radicals is a transformative process that allows for expressions involving powers to be expressed as roots. Any expression with a fractional exponent can be represented in radical form, making it easier to interpret and solve.

Let's look at the process of converting using a formula:

• For an expression like \(x^{n/m}\), rewrite it as \(\sqrt[m]{x^n}\).
• Alternatively, you can express it as \((\sqrt[m]{x})^n\).

This dual representation provides flexibility in mathematical calculations. Following this approach, an expression like \(x^{3/2}\) converts into a radical as \(\sqrt{x}^3\). This expression indicates that \(x\) is first square-rooted and then raised to the third power.

Understanding the conversion from exponents to radicals is key in algebra, providing a basis for solving polynomial equations and other algebraic expressions efficiently.