Problem 12 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression $$ 2^{-1.5} $$

Step-by-Step Solution

Answer

\(2^{-1.5} = \frac{\sqrt{2}}{4}\).

Step 1: Understanding the Expression

The exercise asks us to rewrite the given exponential expression in a different form. We need to interpret \(2^{-1.5}\) using radical notation.

Step 2: Expressing the Exponent in Fraction Form

Start by expressing the exponent \(-1.5\) as a fraction. The fraction form of \(-1.5\) is \(-\frac{3}{2}\). This helps in converting the expression to involve a radical.

Step 3: Rewrite Using the Radical Form

Using the rules of exponents, any base to a rational exponent can be written as a radical. The expression \(2^{-\frac{3}{2}}\) can be expressed as \(\frac{1}{\sqrt[2]{2^3}}\) or simplified further to \(\frac{1}{\sqrt{8}}\).

Step 4: Simplifying the Radical Expression

Calculate \(\sqrt{8}\), which can be simplified to \(2\sqrt{2}\). Therefore, \(\frac{1}{\sqrt{8}}\) further simplifies to \(\frac{1}{2\sqrt{2}}\). To make it more standard, we can rationalize the denominator, resulting \(\frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}\).

Key Concepts

Exponential Expressions

In mathematics, exponential expressions are terms where a number, known as the base, is raised to a power, which is the exponent. For example, in the expression \(2^{-1.5}\), the number 2 is the base, and \(-1.5\) is the exponent. The exponent tells you how many times to multiply the base by itself. When the exponent is negative, as in \(2^{-1.5}\), it signifies the reciprocal of the base raised to the positive of that exponent.

Understanding how to manipulate these expressions is crucial, especially in algebra and calculus. Here’s a quick breakdown:

• A positive exponent like in \(2^3\) means multiply 2 by itself twice (\(2 \times 2 \times 2 = 8\)).
• A negative exponent like in \(2^{-3}\) means take the reciprocal of the positive exponent (\(\frac{1}{2^3} = \frac{1}{8}\)).
• A fractional exponent like in \(2^{0.5}\) is equivalent to the square root of 2; \(2^{0.5} = \sqrt{2}\).

Connecting these pieces helps unlock a deeper understanding of different forms of the same numerical values, offering more flexibility in mathematical expressions.

Rational Exponents

Rational exponents may seem intimidating at first, but they are simply another way of expressing roots using exponents. When you encounter an expression like \(2^{\frac{3}{2}}\), it represents both an exponent and a root. The numerator (3 in this case) tells you the power to which the base should be raised, while the denominator (2 here) signifies the root to be taken.

This converts easily into radical form following these steps:

• Start by expressing the exponent as a fraction if it isn't already. For example, \(2^{-1.5}\) can be written as \(2^{-\frac{3}{2}}\).
• The fraction \(-\frac{3}{2}\) indicates you need both the cube (raise to the third power) and the square root of the base.
• In radical notation, this becomes \(\sqrt[2]{2^3}\) or in a more simplified format \(\sqrt{8}\).

Understanding the fractional exponent provides the base for transforming expressions between exponential and radical forms, which is an essential skill in algebra.

Simplifying Radicals

Simplifying radicals refers to the process of making a radical expression as simple as possible. For example, when dealing with \(\sqrt{8}\), you want to express it in its simplest terms. Here's how you can do this:

• Break down the number under the radical into its prime factors: \(8 = 2 \times 2 \times 2\).
• Pair the factors to simplify: Since \(8\) is \(2^3\), this simplifies to \(2\sqrt{2}\), because \(\sqrt{2^2} = 2\).
• If applicable, rationalize the denominator to remove any radicals from it. This is often done by multiplying the numerator and the denominator by a radical that will eliminate the radical present in the denominator. In this case, you may end up transforming \(\frac{1}{2\sqrt{2}}\) into \(\frac{\sqrt{2}}{4}\).

With these steps, you transition a more complex radical expression into a simpler, easier-to-understand form, vital for solving equations efficiently. This skill streamlines both learning and application of mathematical concepts in higher-level math courses.