Problem 43 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt[5]{2^{3}} $$

Step-by-Step Solution

Answer

The expression \( \sqrt[5]{2^3} \) is \( 2^{3/5} \) in simplest form.

Step 1: Understanding the Radical

In this expression, \( \sqrt[5]{2^3} \), the 5 is the root indicator, and 2 is raised to the power of 3 inside the radical.

Step 2: Converting Radical to Exponent

To convert the radical expression \( \sqrt[5]{2^3} \) to an expression with exponents, use the rule \( \sqrt[n]{a^m} = a^{m/n} \).

Step 3: Applying the Rule

Apply the rule from Step 2: \( 2^{3/5} \).

Step 4: Simplifying (If Needed)

Since \( 2^{3/5} \) cannot be simplified further in terms of the exponents, it is already in its simplest form.

Key Concepts

Radicals to Exponents

Radicals and exponents are closely connected, allowing us to transform one into the other. This transformation is particularly useful in algebra for simplifying complex expressions. A radical expression, such as a root, can be expressed as an exponent. For example, the radical expression \( \sqrt[n]{a^m} \) uses both a root and a power. The key to converting radicals into exponents is understanding their relationship. The root becomes the denominator and the power the numerator. Hence, \( \sqrt[n]{a^m} \) can be expressed as \( a^{m/n} \). This transformation helps simplify calculations and is often used to express final answers in a more conventional algebraic form.

• The radical sign \( \sqrt{} \) indicates a root. The number inside is the radicand.
• For a fifth root \( \sqrt[5]{a^m} \), you would write the expression as \( a^{m/5} \).

Converting radical expressions into exponents often streamlines algebraic processes and aids in carrying out further operations with ease.

Simplifying Exponents

Simplifying expressions with exponents requires understanding their basic properties. Once a radical is converted into an exponential form, like \(2^{3/5}\), you should simplify if possible. Simplifying means reducing the expression to its simplest form or combining terms to make the expression as concise as possible.
Begin by assessing whether there are like terms that can be combined or exponents that can be adjusted. In our example, \(2^{3/5}\) is already in simplest form because the fraction \(\frac{3}{5}\) does not reduce further. Using common properties of exponents can aid in the simplification process:

• When multiplying similar bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• When dividing similar bases, you subtract exponents: \(a^m / a^n = a^{m-n}\).
• When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

By following these basic operations, you can simplify many exponents effectively.

Positive Exponents

Working with positive exponents is fundamental in algebra. A positive exponent indicates how many times to multiply the base by itself. For instance, \(2^3\) means you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\). When writing expressions with positive exponents, it's essential to ensure that there are no negative or fractional exponents unless required.
In our context of simplifying radicals to exponents, the goal typically involves ensuring that the exponents are positive after any conversions. For example, after converting a radical to an exponent, you might end up with a fractional exponent like \(2^{3/5}\). This notation implies that the base 2 is raised to the positive power of three-fifths.

• Positive exponents simplify comparisons and operations, as they follow intuitive multiplying principles.
• Positive exponents eliminate the need for additional steps or inversions that negative exponents often require.

Mastering positive exponents can significantly enhance your ability to handle algebraic expressions and understand the properties behind them.