Problem 102 - Algebra and Trigonometry

Question

Simplify by reducing the index of the radical. $$ \sqrt[4]{7^{2}} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[4]{7^{2}}\) is \(\sqrt{7}\).

Step 1: Translating the Radical

Transform the fourth root of a square into an exponent using the property \(\sqrt[n]{a^{m}} = a^{m/n}\). This translates \(\sqrt[4]{7^{2}}\) into the form \(7^{2/4}\).

Step 2: Simplify the Exponents

Further simplify the exponents by dividing 2 by 4 to get 0.5. Hence, the final form of the equation becomes \(7^{0.5}\).

Step 3: Final Simplification

Express \(7^{0.5}\) in radical form since that is more simplified in this case. Thus \(7^{0.5}\) becomes \(\sqrt{7}\).

Key Concepts

Rational Exponents

When working with rational exponents, understanding the conversion between radicals and exponents is key. Rational exponents are those in the form of fractions, where the numerator represents the power and the denominator indicates the root. For example, the expression \(a^{m/n}\) indicates that the base \(a\) is raised to the power \(m\) and then the \(n\)-th root is taken.

• The expression \(a^{1/n}\) is equivalent to the \(n\)-th root of \(a\).
• \(a^{m/n}\) is equivalent to \(\sqrt[n]{a^m}\).

This is why in the original exercise, \(\sqrt[4]{7^2}\) can be rewritten as \(7^{2/4}\). This interchangeability helps when simplifying expressions and solving equations.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are often rewritten using rational exponents for easier manipulation. The index of a radical specifies the degree of the root. For example, in \(\sqrt[4]{7^2}\), 4 is the index showing a fourth root is required.

• Redefining radicals into rational exponents makes it simpler to apply exponentiation rules.
• Use rational exponents to simplify difficult radical expressions to uncover more straightforward forms.

In this exercise, converting \(\sqrt[4]{7^2}\) to \(7^{2/4}\) allows for simplifying the fraction \(\frac{2}{4}\) to get \(7^{0.5}\). Ultimately, returning to radical form, \(7^{0.5}\) equals \(\sqrt{7}\), a simple radical expression.

Exponentiation Rules

Exponentiation rules facilitate simplifying expressions with exponents. These rules are particularly beneficial when transitioning between radical and exponential forms. Key exponentiation rules include:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power Rule: \((a^m)^n = a^{m \times n}\)
• Rational Exponent Rule: \(a^{m/n} = \sqrt[n]{a^m}\)

For the problem at hand, the Rational Exponent Rule was vital in transforming the fourth root \(\sqrt[4]{7^2}\) to \(7^{2/4}\), allowing us to manipulate it more straightforwardly. Understanding these rules and how they interconnect helps in easing complex problems into simpler, more manageable parts.