Question

\(\left(12^2\right)^{1 / 4}\)

Short answer

The simplified and final value of the original expression \(\left(12^2\right)^{1 / 4}\) is \(3\).

Step-by-step solution

Write Out the Base and Exponent

At first, simplify the base to get the actual number. Here, the base number is \(12^2\). Square \(12\) to get \(144\).

Apply the Power Rule

Now consider \(\left(144\right)^{1 / 4}\). Applying the power rule here, we take the fourth root of \(144\) which yields \(\sqrt[4]{144} \).

Calculate the Root

Calculate the fourth root of \(144\), which gives us \(3\).

Key concepts

Power Rule

When we work with exponents, it is essential to understand the power rule as it simplifies the process of dealing with complex expressions. The power rule is a fundamental concept in mathematics that describes what happens when you raise a power to another power. The rule states that to simplify an expression of the form \( (a^m)^n \) , you multiply the exponents together to get \( a^{m \times n} \) .

Let's apply this to the original exercise \( \big(12^2\big)^{1/4} \) . Here, according to the power rule, we would multiply the exponents: \( 2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \) . As a result, the expression simplifies to \( 12^{1/2} \) , which represents the square root of 12. However, in the provided step by step solution, they first calculate \( 12^2 \) and then apply the fourth root, which also leads to the correct result but is a bit longer process.

The Importance of the Power Rule

Understanding the power rule is crucial because it helps you simplify expressions without calculating large numbers and allows for easier manipulation of exponents in algebraic equations. It is a core tool when working with exponential functions and simplifying expressions involving powers.

Simplifying Expressions

Simplifying expressions is a process that makes mathematical expressions easier to work with by reducing them to their simplest form. This does not change the value of the expression; it just makes it easier to read and understand. When simplifying expressions, you combine like terms, use algebraic properties, such as the distributive property, and apply rules of exponents like the power rule.

For instance, in the exercise \( \big(12^2\big)^{1/4} \) , one can simplify the base first to get \( 144 \) and then simplify further by taking the fourth root. This step-by-step approach allows for a structured method of simplifying an expression and ensures that none of the properties of numbers and operations are violated.

Tips for Simplifying Expressions

There are several strategies to simplify expressions effectively:

• Combine like terms
• Factor out common factors
• Use the power rule for exponents
• Recognize and apply special algebraic formulas

By following these strategies, students can efficiently tackle complex algebraic expressions and prevent common errors.

Roots of Numbers

The roots of numbers are inverse operations to exponentiation. When we talk about roots, the most common are the square root and the cube root, but roots can be of any degree. The n-th root of a number x is written as \( \root{n}oints{x} \) and it represents the number which when raised to the power of n gives x. For example, the square root of 9 is 3 because \( 3^2 = 9 \) , and, similarly, the fourth root of 16 is 2 because \( 2^4 = 16 \) .

In the context of our exercise, calculating the fourth root of 144 is required, which means finding a number that when raised to the power of 4 gives 144. This number is 3 because \( 3^4 = 81 \times 3 = 243 \times 3 = 729 \times 3 = 2187 \) . However, there is an error in the step-by-step solution as it states that the fourth root of 144 is 3, whereas it's actually 12. Recognizing and correcting such errors is an important part of understanding roots.

Dealing with Radicals

Radicals can sometimes seem daunting, but with practice, they become manageable. Whether you are simplifying square roots or higher order roots, the process usually involves prime factorization or recognizing perfect powers within the number you're working with. Simplifying radicals also often involves rationalizing denominators when the radical is in the denominator of a fraction.