Problem 538 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(16^{\frac{1}{4}}\) (b) \(16^{\frac{1}{2}}\) (c) \(3125^{\frac{1}{5}}\)

Step-by-Step Solution

Answer

(a) 2 (b) 4 (c) 5

Step 1: Identify the Expression

There are three parts to this exercise: (a) simplify \(16^{\frac{1}{4}}\), (b) simplify \(16^{\frac{1}{2}}\), and (c) simplify \(3125^{\frac{1}{5}}\).

Step 1: Simplify \(16^{\frac{1}{4}}\)

The fraction \(\frac{1}{4}\) signifies the fourth root of 16. We can rewrite the expression as \(\sqrt[4]{16}\). We know that \(2^4 = 16\), so \(\sqrt[4]{16} = 2\). Hence, \(16^{\frac{1}{4}} = 2\).

Step 2: Simplify \(16^{\frac{1}{2}}\)

The fraction \(\frac{1}{2}\) signifies the square root of 16. We can rewrite the expression as \(\sqrt{16}\). Since \(4^2 = 16\), \(\sqrt{16} = 4\). Therefore, \(16^{\frac{1}{2}} = 4\).

Step 3: Simplify \(3125^{\frac{1}{5}}\)

The fraction \(\frac{1}{5}\) signifies the fifth root of 3125. We can rewrite the expression as \(\sqrt[5]{3125}\). Knowing that \(5^5 = 3125\), \(\sqrt[5]{3125} = 5\). Thus, \(3125^{\frac{1}{5}} = 5\).

Key Concepts

fractional exponents

In mathematics, fractional exponents are a way to represent roots using exponentiation. A fractional exponent like \(\frac{1}{n}\) means you are taking the \(n\)-th root of a number. For example, \(16^{\frac{1}{4}}\) means finding the fourth root of 16.

To transform a number to its fractional exponent form, you can use the equivalence: \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). In simpler terms:

• The numerator (the \(m\)) represents the standard exponentiation (raising to a power).
• The denominator (the \(\frac{1}{n}\)) represents the root (for example, square root, cube root).

Fractional exponents provide a powerful and efficient way to represent and compute roots. Instead of writing \(\frac{1}{5}\)-root of 3125 as \( \text{fifth root of } 3125\) it is more succinctly written as \(\text{3125}^{\frac{1}{5}}\).

roots of numbers

Roots of numbers are one of the foundational concepts in mathematics. A root can be thought of as the opposite of raising a number to a power. For example, the square root of 16 is the number that, when multiplied by itself, results in 16.

Different roots include:

• Square roots: This is the second root and is written using the symbol \(\sqrt{...}\). An example is \(\sqrt{16} = 4\) because \(4^2 = 16\).
• Cube roots: This is the third root and is written with the cube root symbol \(\sqrt[3]{...}\). An example is \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).
• Higher-order roots: These include the fourth root, fifth root, and so on, and are written as \(\sqrt[n]{...}\) where \(n\) indicates the root order.

To relate it to the original exercise:

• \(16^{\frac{1}{4}}\) was simplified because \(\sqrt[4]{16} = 2\).
• \(16^{\frac{1}{2}}\) can be explained with square root, \(\sqrt{16} = 4\).
• \(3125^{\frac{1}{5}}\) means \(\sqrt[5]{3125} = 5\).

exponentiation

Exponentiation is a mathematical operation written as \(a^b\), involving two numbers (the base \(a\) and the exponent or power \(b\)). This means that \(a\) is multiplied by itself \(b\) times. For instance, \(2^3\) means \(2 \times 2\times 2 = 8\).

The main rules of exponentiation are:

• \(a^m \times a^n = a^{m+n}\)
• \(a^m / a^n = a^{m-n}\)
• \((a^m)^n = a^{m \times n}\)
• \(a^{-n} = \frac{1}{a^n}\)

When dealing with fractional exponents, these rules still apply, but the exponents can be fractions. For example:

• \(a^{\frac{1}{4}}\) implies taking the fourth root of \(a\).
• \(a^{\frac{2}{3}}\) implies taking \text{cube root of } a\text{ and then squaring the result.}

Using these rules helps in simplifying expressions that involve both whole and fractional exponents, making complex calculations more manageable.