Question

Simplify the expression.\(\sqrt[4]{\left(3 x^{2}\right)^{4}}\)

Short answer

The simplified expression is \(3x^2\)

Step-by-step solution

Apply the power rule

Applying the power rule, \((a^{m})^{n} = a^{mn}\), the expression \((3x^{2})^{4}\) becomes \((3x^{2})^{4*\frac{1}{4}} = 3^{\frac{4}{4}} x^{2*\frac{4}{4}}\)

Simplify the exponents

Simplify \((3x^{2})^{4*\frac{1}{4}}\). The exponent of \(3\) becomes \(4*\frac{1}{4} = 1\) and the exponent of \(x\) becomes \(2*\frac{4}{4} = 2\). The expression which previously was \((3x^{2})^{4*\frac{1}{4}}\) is now \(3^{1}*x^{2} = 3x^2\)

Final simplification

The simplified version of the original expression \(\sqrt[4]{\left(3 x^{2}\right)^{4}}\) is \(3x^2\)

Key concepts

Exponent Rules

Understanding exponent rules is essential for simplifying algebraic expressions with powers. An exponent, or power, represents the number of times a base number is multiplied by itself. For instance, when we see a term like \( a^m \), it means that the base \( a \) is multiplied by itself \( m \) times. When working with expressions that contain exponents, there are several key rules to remember:

• Product Rule: \( a^m \times a^n = a^{m+n} \) – This means when you multiply two exponential expressions with the same base, you add their exponents.
• Quotient Rule: \( a^m \div a^n = a^{m-n} \) – When dividing two exponential expressions with the same base, subtract the exponents.
• Power of a Power Rule: \( (a^m)^n = a^{mn} \) – When you have an exponent raised to another exponent, you multiply the exponents.
• Zero Exponent Rule: \( a^0 = 1 \) – Any base raised to the zero power equals one.
• Negative Exponent Rule: \( a^{-n} = 1 / a^n \) – A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.

These rules allow for the simplification of complex expressions by manipulating exponents in a logical and consistent manner.

Radicals

Radicals, or root expressions, are another fundamental concept in algebra. A radical can be represented as \( \sqrt[n]{a} \), where \( a \) is the radicand, and \( n \) is the index or degree of the root. Simplifying radicals involves several tactics, including finding perfect nth powers within the radicand and applying rational exponent notation. Rational exponent notation relates radicals to exponents by expressing a root as an exponent fraction, with \( \sqrt[n]{a} = a^{\frac{1}{n}} \). This conversion facilitates applying other exponent rules to simplify the radicals further. For example, when a radical has a power, such as in the provided exercise, we can translate \( \sqrt[4]{(3x^2)^4} \) to \( (3x^2)^{\frac{4}{4}} \), which enables the use of exponent rules to simplify the expression into a more manageable form. Always keep an eye out for opportunities to extract perfect roots when working with radicals to reduce the expression to its simplest form.

Algebraic Manipulation

Algebraic manipulation encompasses a variety of techniques used to rewrite and simplify mathematical expressions. This process often includes factoring, expanding, simplifying fractions, and combining like terms. The goal is to transform an equation or expression into its simplest form or into a form that enables you to solve an equation or perform further operations with ease.

In manipulating algebraic expressions, it's essential to adhere to the basic properties of arithmetic, including the commutative, associative, and distributive properties. Whether you are dealing with exponents, radicals, or polynomial terms, being skilled in algebraic manipulation allows for the simplification of complex problems into more solvable steps, as seen in the problem \( \sqrt[4]{(3x^2)^4} \). By combining the knowledge of exponent rules and an understanding of how to appropriately apply them, we can significantly simplify the initial complex expression.

Power Rule

The power rule is a specific exponent rule used when an exponent is raised to another exponent. This situation occurs quite commonly in algebra and calculus. The rule states that for any expression of the form \( (a^m)^n \), you can multiply the exponents to simplify the power, like so: \( (a^m)^n = a^{mn} \). This simplification is especially powerful because it often eliminates the need for complex computations involving large numbers.

For instance, in our example of \( \sqrt[4]{(3x^2)^4} \), applying the power rule is the first step. Translating the radical to rational exponents, we get \( (3x^2)^{4*\frac{1}{4}} \). According to the power rule, we multiply the exponents, resulting in \( (3x^2)^{1} \), which simplifies to \( 3x^2 \). This fundamental rule is essential for streamlining calculations and solving algebraic problems involving powers and roots more efficiently.