In Exercises 21–38, rewrite each expression with rational exponents. _ √7

Verified step by step guidance

1. Identify the expression given: \( \sqrt{7} \).
2. Recall that a square root can be expressed as a rational exponent. The square root of a number \( a \) is equivalent to \( a^{1/2} \).
3. Apply this rule to the expression: \( \sqrt{7} = 7^{1/2} \).
4. The expression \( 7^{1/2} \) is the rewritten form of \( \sqrt{7} \) using rational exponents.
5. Verify that the expression is now in the form of a rational exponent, which is \( 7^{1/2} \).

Key Concepts

Rational Exponents

Rational exponents are exponents that can be expressed as a fraction, where the numerator indicates the power and the denominator indicates the root. For example, an exponent of 1/2 represents the square root, while 1/3 represents the cube root. This notation allows for a more compact representation of roots and powers, facilitating easier manipulation of expressions in algebra.

Radical Notation

Radical notation is a way to express roots using the radical symbol (√). For instance, √a denotes the square root of 'a'. Understanding how to convert between radical notation and rational exponents is crucial, as it allows for the simplification and rewriting of expressions in different forms, which is often required in algebraic operations.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving exponents. Key properties include the product of powers, quotient of powers, and power of a power. These properties are essential for simplifying expressions and solving equations, especially when converting between radical and rational exponent forms.