Question

Rewrite the rational exponent as a radical by extending the properties of integer exponents.

Answer 1

The rational exponent
`\(a^(m/n)\)` can be expressed as the nth root of a raised to the power of m, where a is the base, m is the numerator, and n is the denominator.

Step-by-step explanation:

Rational exponents, expressed in the form
`\(a^(m/n)\),`can be rewritten as radicals by applying the properties of integer exponents. The exponent m/n signifies the power to which the base a is raised. To express this as a radical, we extend the concept of radicals to include fractional exponents.

In mathematical terms,
`\(a^(m/n)\)`is equivalent to the nth root of a raised to the power of m. This can be denoted as
`\(\sqrt[n]{a^m}\).`Here,
`\(\sqrt[n]{a}\)`represents the nth root of a). The numerator m indicates the power to which the nth root of a is raised. This transformation allows us to represent rational exponents in a more familiar radical form.

For example, if we have
`\(2^(3/2)\),` we can rewrite this as
`\(\sqrt[2]{2^3}\).` This translates to the square root of 2 cubed, yielding the same result as the original rational exponent. By employing this method, we bridge the gap between rational exponents and radicals, providing a clearer and more intuitive representation of the mathematical expression.