Question

Simplify the expression.\(\frac{5^{1 / 2}}{5}\)

Short answer

The simplified expression of \(\frac{5^{1 / 2}}{5}\) is \(1 / \sqrt{5}\).

Step-by-step solution

Express the denominator in exponential form

In our case, the number 5 in the denominator can be rewritten as \(5^{1}\) since any number raised to the power of 1 remains the same.

Use the rule of exponents for division

According to the rule of exponents for division, if you are dividing like bases then you subtract the exponents. Using this rule, the expression becomes \(5^{1 / 2 - 1}\).

Simplify the exponent

Subtract the fractions in the exponent to simplify further. \(1 / 2 - 1 = -1 / 2\). So, the expression simplifies to \(5^{-1 / 2}\).

Rewrite as positive exponent

A negative exponent means that the base is on the wrong side of the fraction line, so you flip the base to the other side. When we remove the negative on the exponent, the expression becomes \(\frac{1}{5^{1 / 2}}\) or \(1 / \sqrt{5}\) when rewritten without exponents.

Key concepts

Exponent Rules

Understanding exponent rules is crucial for simplifying exponential expressions. One fundamental rule is the division rule, which states that when you divide exponential expressions with the same base, you can subtract the exponents. In mathematical terms, if you have a base number a, raised to two different powers, m and n, the division is expressed as a^m / aⁿ = a^m-n. This rule greatly simplifies the process of division when dealing with exponential terms.

In the step-by-step solution provided, we applied this rule to the expression 5^1/2 / 5, which simplified to 5^{1/2 - 1}.

Negative Exponents

Negative exponents often confuse students, but the concept is quite straightforward. A negative exponent indicates that the base should be inverted and the exponent made positive. The general rule is that a^-n = 1 / aⁿ, meaning that you take the reciprocal of the base raised to the positive exponent.

For example, in the given exercise, once we arrived at the expression 5^-1/2, we recognized that a negative exponent needs to be converted. Thus, the expression transformed into 1 / 5^1/2, which is the correct representation with a positive exponent.

Algebraic Fractions

Algebraic fractions are fractions that contain algebraic expressions in the numerator, the denominator, or both. Simplifying algebraic fractions involves finding a way to make the expressions as straightforward as possible often by factoring, reducing, and cancelling terms where applicable.

When exponential expressions are part of algebraic fractions, as in the textbook problem, exponent rules, including those for negative exponents, are used. Once we simplified the expression in the example to 1 / 5^1/2, rewriting it without exponents gives us 1 / √5. In some cases, we might further rationalize the denominator (a process not needed here), which involves eliminating the radical from the denominator, thereby simplifying the algebraic fraction further.