In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible.49^-½

Verified step by step guidance

1. Identify the expression given: \( 49^{-\frac{1}{2}} \).
2. Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent. So, \( a^{-n} = \frac{1}{a^n} \).
3. Apply this rule to the expression: \( 49^{-\frac{1}{2}} = \frac{1}{49^{\frac{1}{2}}} \).
4. Recognize that an exponent of \( \frac{1}{2} \) represents the square root. Therefore, \( 49^{\frac{1}{2}} = \sqrt{49} \).
5. Substitute back into the expression: \( \frac{1}{\sqrt{49}} \).

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 corresponds to the square root, while an exponent of 1/3 corresponds to the cube root. Understanding how to convert between radical expressions and rational exponents is essential for simplifying expressions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For instance, a term like a^-n can be rewritten as 1/(a^n). This concept is crucial for rewriting expressions with negative exponents into a more manageable form, particularly when simplifying or rewriting expressions with positive rational exponents.

Simplification of Expressions

Simplification involves rewriting an expression in a more concise or manageable form, often by combining like terms, reducing fractions, or applying exponent rules. In the context of rational exponents, this may include converting negative exponents to positive ones and simplifying any resulting radical expressions. Mastery of simplification techniques is vital for effectively solving algebraic problems.