Simplify by reducing the index of the radical. 724\sqrt[4]{$$7^2$$}

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 102

1장, 문제 102

Simplify by reducing the index of the radical. $the fourth root of 7 squared$

검증된 단계별 안내

Identify the expression inside the radical: the fourth root of $7^2$, which can be written as $\sqrt[4]{7^2}$.

Recall the property of radicals and exponents: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Apply this to rewrite the expression as $7^{\frac{2}{4}}$.

Simplify the fraction in the exponent: $\frac{2}{4} = \frac{1}{2}$, so the expression becomes $7^{\frac{1}{2}}$.

Recognize that $7^{\frac{1}{2}}$ is equivalent to the square root of 7, or $\sqrt{7}$.

Thus, the simplified form of $\sqrt[4]{7^2}$ is $\sqrt{7}$.

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주요 개념

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Radical Expressions and Indices

A radical expression involves roots, such as square roots or fourth roots, indicated by an index. The index shows the degree of the root, for example, the fourth root (⁴√) means the number raised to the 1/4 power. Understanding how to interpret and manipulate these indices is essential for simplifying radicals.

Reducing the Index of a Radical

Reducing the index means rewriting a radical with a smaller root index or converting it into an equivalent expression with fractional exponents. This often involves expressing the radicand with powers that allow simplification, such as rewriting ⁴√(7²) as 7^(2/4) and then simplifying the fraction.

Fractional Exponents and Simplification

Fractional exponents represent roots and powers simultaneously, where a^(m/n) equals the nth root of a raised to the mth power. Simplifying expressions with fractional exponents involves reducing the fraction and rewriting the expression in simplest form, which helps in reducing the index of radicals effectively.

Simplify by reducing the index of the radical. 724\(\sqrt\)[4]{7^2}