Problem 23 - Contemporary Precalculus

Question

Write the given expression without using radicals. $$\frac{1}{\sqrt{x}}$$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\frac{1}{\sqrt{x}}\) without using radicals. Answer: \(x^{-\frac{1}{2}}\)

Step 1: Rewrite the square root as an exponent

We rewrite the square root of \(x\) as \(x^{\frac{1}{2}}\). The given expression will be: $$\frac{1}{x^{\frac{1}{2}}}$$

Step 2: Simplify the expression

Now, bring the \(x^{\frac{1}{2}}\) to the numerator to remove the fraction in the denominator. The exponent would become negative: $$x^{-\frac{1}{2}}$$ Now, the expression is written without using radicals.

Key Concepts

Understanding Square Roots

In understanding mathematics, the concept of a square root is fundamental. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because when 3 is multiplied by 3, you get 9. In algebra, the square root of a variable like x is represented as \(\sqrt{x}\). This can be challenging when dealing with complex expressions, but a grasp of this concept is crucial.

One point where students often stumble is recognizing that square roots can be represented as fractional exponents. Understanding this property allows for a more flexible manipulation of expressions, especially when variables are involved or when the expressions are part of a larger equation. When you see a square root, remember it's an operation asking 'what number, when squared, gives me the original number?'

Exponentiation: Power to the Numbers

The operation of raising a number to a power is known as exponentiation. The exponent represents the number of times a base number is multiplied by itself. For instance, \(2^3\) means 2 is multiplied by itself 3 times: 2 × 2 × 2, resulting in 8. The base here is 2, and the exponent is 3. Exponentiation is a shortcut for repeated multiplication and a fundamental tool in algebra.

Working With Negative Exponents

Students may be confused by negative exponents. It's simpler than it seems: a negative exponent means you take the reciprocal of the base raised to the absolute value of the exponent. For example, \(x^{-2}\) is the same as \(\frac{1}{x^2}\). This concept extends to fractional exponents as well. Always remember, exponents are not there to complicate the problem but to simplify the process of multiplication.

Rational Exponents: Fractions Get Powerful

A rational exponent means an exponent that is a fraction. For example, the expression \(x^{\frac{1}{2}}\) uses a rational exponent, which is another way to denote the square root of x. This notation is particularly useful in algebra because it allows the use of exponent rules to simplify expressions with roots.

How Rational Exponents Simplify Expressions

Rational exponents make it easier to perform operations such as multiplication and division with roots. For instance, \(\sqrt{x}\) and \(x^{\frac{1}{2}}\) are equivalent - they both represent the principal square root of x. By expressing square roots as rational exponents, we can apply the laws of exponents to simplify complex expressions, which might be difficult to manage with radical symbols. These conversions are a key part of solving algebraic problems, especially when dealing with higher-level equations or when integrating other algebraic rules.