Question

Use rational exponents to reduce the index of the radical.\(\sqrt{\sqrt{32}}\)

Short answer

The simplified expression of \(\sqrt{\sqrt{32}}\) is 2.

Step-by-step solution

Rewrite the radical using rational exponents

A square root can be represented as an exponent of 1/2. Thus, \(\sqrt{32}\) can be written as \(32^{1/2}\). And again the square root of that which is \(\sqrt{\sqrt{32}}\) can be written as \((32^{1/2})^{1/2}\)

Simplify the exponent

According to the rules of exponents, we can multiply the exponents when an exponent is raised to another exponent. So, \((32^{1/2})^{1/2}\) becomes \(32^{1/2×1/2}\) which simplifies to \(32^{1/4}\).

Simplify the expression

Now, we compute the fourth root of 32. This is equal to 2. So, the simplified expression is equal to 2.

Key concepts

Radical Expressions

Radical expressions often involve roots, such as square roots or cube roots. In mathematics, roots can be represented using the radical sign, like \( \sqrt{} \), or via fractional exponents, also known as rational exponents.
For example, the square root of a number \(a\) is written as \( \sqrt{a} \) or \( a^{1/2} \).
Converting a radical expression into a form with rational exponents can simplify complex calculations. It allows us to leverage **exponent rules** more effectively in any algebraic manipulation. You can break down radical expressions with higher indices by rewriting them using rational exponents, helping you to systematically solve for values and simplify expressions.

Exponent Rules

Exponent rules are guidelines that ease the simplification of expressions involving powers. These rules make it possible to multiply, divide, and power numbers efficiently. Understanding exponent rules is crucial when working with rational exponents because it simplifies handling expressions mathematically. Some key rules include:

• \( a^m \times a^n = a^{m+n} \): This rule helps in multiplying powers with the same base by adding their exponents.
• \( (a^m)^n = a^{m\cdot n} \): When taking powers of powers, multiply the exponents together.
• \( a^{-n} = \frac{1}{a^n} \): A negative exponent means you take the reciprocal of the base.

In our original exercise, the rule \((a^m)^n = a^{m\cdot n}\) was particularly useful to simplify the nested square roots into a single expression of \( 32^{1/4} \). Clearly applying these rules allows more straightforward and clear computation of complex expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise form without changing their value. This process often uses exponent rules, especially when working with rational exponents.
The ultimate goal is to express the concept in its simplest form, making it easier to understand and work with.
Let's consider our task: \( \sqrt{\sqrt{32}} \). By expressing each square root as a power, we simplified it step by step.

• First, express the square root of 32 as a power: \( 32^{1/2} \).
• Second, simplify the nested expression \((32^{1/2})^{1/2}\) using exponent rules: it becomes \(32^{1/4}\).
• Finally, compute the value, which simplifies directly to 2.

Approaching problems in smaller steps not only simplifies the math but also clarifies each transformation in the solution process.