Problem 5 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ \left(\frac{1}{16}\right)^{1 / 4} $$

Step-by-Step Solution

Answer

The radical notation is \( \sqrt[4]{\frac{1}{16}} \), simplified to \( \frac{1}{2} \).

Step 1: Understand the Problem

We need to rewrite the expression \( \left(\frac{1}{16}\right)^{1/4} \) using radical notation and simplify it if possible.

Step 2: Convert to Radical Notation

The expression \( a^{1/n} \) is equal to \( \sqrt[n]{a} \). Therefore, \( \left(\frac{1}{16}\right)^{1/4} \) can be rewritten as \( \sqrt[4]{\frac{1}{16}} \).

Step 3: Simplify the Radical

The expression \( \sqrt[4]{\frac{1}{16}} \) can be simplified. Since \(16 = 2^4\), \( \frac{1}{16} \) can be written as \( \left(2^{-4}\right) \). Therefore, \( \sqrt[4]{\frac{1}{16}} = \left(2^{-4}\right)^{1/4} = 2^{-1} = \frac{1}{2} \).

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in mathematics that allow us to express repeated multiplication concisely. For instance, when we have a number like 2 multiplied by itself 4 times (i.e., \(2 \times 2 \times 2 \times 2\)), it is written as \(2^4\). Here, 2 is the base, and 4 is the exponent, meaning "2 raised to the power of 4". This notation simplifies expressions, especially when dealing with large numbers or complex calculations.
To better grasp exponents, remember:

• The base is the number being multiplied.
• The exponent indicates how many times the base is used as a factor.

Understanding exponents is essential because they are the stepping stone to more advanced topics such as roots and rational exponents.

Simplifying Expressions with Exponents

Simplifying expressions makes them easier to work with and understand. When simplifying expressions with exponents, a few rules can help:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: \((a^m)^n = a^{m \cdot n}\).
• Power of a Product: \((ab)^n = a^n \cdot b^n\).
• Power of a Fraction: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \).

When dealing with negative bases or exponents, be extra cautious: - A negative base raised to an even power results in a positive number, while an odd power keeps the base negative.- A negative exponent signifies taking the reciprocal of the base, as seen in \(a^{-n} = \frac{1}{a^n}\).In the provided exercise, we used the fact that \(16\) is \(2^4\), allowing us to simplify \(\sqrt[4]{\frac{1}{16}}\) to \(\frac{1}{2}\). This rule helps in breaking down complex expressions into simpler ones.

Delving into Rational Exponents

Rational exponents are exponents that are fractions. They serve as a bridge between integer exponents and roots. For example, an expression like \(a^{\frac{1}{n}}\) translates to the nth root of \(a\), written as \(\sqrt[n]{a}\).
Rational exponents follow rules similar to those of whole-number exponents:

• \(a^{m/n} = \sqrt[n]{a^m}\) or \((\sqrt[n]{a})^m\).
• Converting between radical notation and rational exponents can make it easier to apply algebraic rules.

Understanding rational exponents is crucial when you need to simplify complex expressions, especially those involving roots and fractional powers. In the original exercise, \(\left(\frac{1}{16}\right)^{1/4}\) means taking the fourth root of \(\frac{1}{16}\), which we simplified to \(\frac{1}{2}\). This conversion often simplifies computation and solves complex equations more efficiently.