Problem 30 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[4]{0.0256} $$

Step-by-Step Solution

Answer

The value of \( \sqrt[4]{0.0256} \) is 0.4.

Step 1: Understand the Expression

The expression we need to evaluate is \( \sqrt[4]{0.0256} \). This represents the fourth root of 0.0256. In simpler terms, we need to find a number which, when multiplied by itself four times, gives 0.0256.

Step 2: Express as a Power

The fourth root can also be expressed using exponents. Recall that \( \sqrt[4]{a} = a^{1/4} \). Thus, \( \sqrt[4]{0.0256} = 0.0256^{1/4} \). Our task is to find \( 0.0256^{1/4} \).

Step 3: Simplify the Base

First, recognize that 0.0256 can be rewritten as \( \frac{256}{10000} \). This simplifies to \( \left(\frac{16}{100}\right)^2 \) since \( 256 = 16^2 \) and \( 10000 = 100^2 \).

Step 4: Apply the Laws of Exponents

Using the property \( (a^m)^n = a^{mn} \), we can write:\[ \left( \frac{16}{100} \right)^2 = \left( \frac{16}{100} \right)^{2 \times \frac{1}{4}} = \left( \frac{16}{100} \right)^{\frac{1}{2}} \].

Step 5: Calculate the Square Root

Now calculate \( \left( \frac{16}{100} \right)^{1/2} \). This further simplifies to \( \frac{16^{1/2}}{100^{1/2}} \), or \( \frac{4}{10} \).

Step 6: Simplify

Finally, simplify \( \frac{4}{10} \) to \( 0.4 \). Therefore, \( \sqrt[4]{0.0256} = 0.4 \).

Key Concepts

Fourth Roots

The concept of fourth roots can be a bit abstract at first. When we talk about the fourth root of a number, we mean finding a value that, when multiplied by itself four times (i.e., raised to the fourth power), equals the original number. For example, if we want to find \( \sqrt[4]{16} \), we need a number that multiplies to give 16 when used four times. That number is 2, because \( 2 \times 2 \times 2 \times 2 = 16 \).

In the given problem, the expression \( \sqrt[4]{0.0256} \) requires us to determine what number, when raised to the power of four, results in 0.0256. This involves understanding how roots and powers work together in various mathematical calculations, making fourth roots a slightly more complex version compared to square roots.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number. For example, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \). The number 3 is the base, and 4 is the exponent. In our exercise, we see exponents used to express the fourth root: \( 0.0256^{1/4} \).

This transformation from a root to an exponent form is powerful because it allows us to use various exponent rules to simplify complex expressions. Understanding how to manipulate exponents is crucial in further simplifying and solving mathematical problems, especially when dividing or combining different powers. It makes dealing with tiny decimals like 0.0256 much more manageable when expressed as an exponent.

Simplification

Simplification is about breaking down complex expressions into their simplest forms. In the solution to \( \sqrt[4]{0.0256} \), simplification plays a major role. Initially, 0.0256 can be kind of intimidating to work with, but by expressing it as a fraction, the simplification becomes more straightforward. The decimal 0.0256 can be written as \( \frac{256}{10000} \).

This fraction is further simplified by recognizing that \( 256 \) is \( 16^2 \) and \( 10000 \) is \( 100^2 \). By doing this, we rewrite the expression to \( \left( \frac{16}{100} \right)^2 \), making it easier to apply exponent rules and eventually calculate its fourth root. Simplification helps reduce errors and makes complex problems more approachable.

Laws of Exponents

The laws of exponents are essential rules that allow us to simplify and manipulate exponential expressions. Here are some key laws:

• \( (a^m)^n = a^{m\cdot n} \) — an exponent raised to another exponent.
• \( a^m \times a^n = a^{m+n} \) — multiplying powers with the same base.
• \( \frac{a^m}{a^n} = a^{m-n} \) — dividing powers with the same base.

In the solution for \( \sqrt[4]{0.0256} \), we used the property \( (a^m)^n = a^{mn} \). Starting from \( \left( \frac{16}{100} \right)^2 \), we simplified it using the exponent rule to \( \left( \frac{16}{100} \right)^{\frac{1}{2}} \).

By applying these rules, problems that initially seem complicated become solvable through logical steps, leading to an accurate final result.