nth Roots and Rational Exponents

Front

What is the definition of an nth root?

Back

The nth root of a number x is a number r such that r^n = x. It is denoted as \( \sqrt[n]{x} \) or x^(1/n).

Front

What is the relationship between rational exponents and roots?

Back

A rational exponent \( \frac{m}{n} \) can be expressed as \( \sqrt[n]{x^m} \) or \( (\sqrt[n]{x})^m \).

Front

Simplify \( \sqrt[5]{-32} \).

Back

The answer is -2, since (-2)^5 = -32.

Front

Convert \( \sqrt[5]{-32} \) to exponential form.

Back

The exponential form is \( (-32)^{\frac{1}{5}} \).

Front

What is the radical form of \( x^{\frac{3}{2}} \)?

Back

The radical form is \( \sqrt{x^3} \) or \( \sqrt{x}^3 \).

Front

How do you express \( a^{\frac{m}{n}} \) in radical form?

Back

It can be expressed as \( \sqrt[n]{a^m} \).

Front

What is the value of \( 64^{\frac{1}{3}} \)?

Back

The value is 4, since 4^3 = 64.

Front

Simplify \( \sqrt{16} \).

Back

The answer is 4.

Front

What is the definition of a rational exponent?

Back

A rational exponent is an exponent that is a fraction, representing both a power and a root.

Front

Convert \( 27^{\frac{2}{3}} \) to radical form.

Back

The radical form is \( \sqrt[3]{27^2} \).