Problem 69 - Algebra 2

Question

Simplify each radical expression. \(n\) is an odd number. $$ \sqrt[n]{m^{4 n}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(m^{4}\).

Step 1: Understand the expression

The given radical expression is \(\sqrt[n]{m^{4 n}}\). Here, \(n\) is an odd number and \(m^{4 n}\) is the radicand. To simplify this, we must recognize that it follows the form \(a^{m*n}\) and apply the rule of exponents.

Step 2: Use the exponent rule

We can rewrite \(m^{4 n}\) as \((m^{4})^{n}\). This fits the rule of exponents better and gives us a clearer way to simplify the expression.

Step 3: Simplify the expression

Once we've rewritten the expression as \(\sqrt[n]{(m^{4})^{n}}\), simplifying it becomes easy. The nth root of a number raised to the power of n is just the original number, so the final answer here is \(m^{4}\).

Key Concepts

Exponent Rules

Exponent rules are crucial for simplifying expressions with powers. These rules help us manage and transform expressions with exponents in a straightforward way. Understanding these rules allows us to simplify complex expressions and make mathematical expressions easier to work with. Let's break down some important exponent rules you might use:

• Product of Powers Rule: When you multiply two expressions with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When you raise a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Quotient of Powers Rule: When you divide two expressions with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Zero Exponent Rule: Any base with an exponent of zero is 1: \(a^0 = 1\).

These rules are the building blocks for simplifying more complex expressions such as those involving radicals and nth roots.

Nth Roots

Nth roots are a generalization of square roots. They involve finding a number that, when raised to the nth power, equals the original number. The symbol \(\sqrt[n]{x}\) denotes the nth root of \(x\). Key points about nth roots include:

• Nth roots can be applied to any real number, and the expression \(\sqrt[n]{x}\) represents the solution \(y\) of \(y^n = x\).
• Having an odd value for \(n\) (like in our original problem) guarantees that \(\sqrt[n]{x}\) has the same parity as \(x\), meaning the root will be positive if \(x\) is positive and negative if \(x\) is negative.
• When \(n\) is even, nth roots imply you may handle both positive and negative values of \(x\), leading to multiple results, usually in pairs: a positive and a negative root.

Understanding nth roots helps simplify expressions that require extracting roots from terms raised to powers, especially when the power and the root are the same.

Simplification Techniques

Simplification techniques are a vital part of mathematics, allowing you to reduce expressions into their simplest form efficiently. This not only makes expressions easier to understand but also to manipulate for further calculations. When simplifying radical expressions, consider the following:

• Identify Common Factors: Recognize any common factors across terms in the expression. This helps you factor them out and simplifies the calculation process.
• Utilize Exponent Rules: Make use of exponent rules to rewrite and simplify expressions, especially when they are in the form of powers raised to powers, like \((m^{4})^{n}\).
• Root-Power Cancellation: Utilizing the knowledge that an nth root of a number to the power of n, such as in \(\sqrt[n]{(m^{4})^{n}}\), cancels to give \(m^{4}\).
• Consistent Forms: Rewriting expressions into forms that are easier to simplify is beneficial, such as converting \(m^{4n}\) to \((m^4)^n\) when involving nth roots.

Mastering these techniques allows for a more streamlined calculation process and a clearer understanding of the mathematical principles involved.