Question

In Exercises \(11-16,\) write the expression as an equivalent expression in the form \(x^{n}\) and give the value for \(n\). $$ \frac{1}{\sqrt{x}} $$

Short answer

Question: Rewrite the expression \(\frac{1}{\sqrt{x}}\) in the form \(x^n\), and find the value of \(n\). Answer: The equivalent expression is \(x^{-\frac{1}{2}}\), and the value of \(n\) is \(-\frac{1}{2}\).

Step-by-step solution

Convert the square root to a power

Recall that the square root of a number can also be represented as a fractional exponent. In particular, $$ \sqrt{x} = x^{\frac{1}{2}} $$ So the given expression can be rewritten as $$ \frac{1}{x^{\frac{1}{2}}} $$

Simplify the expression using properties of exponents

The next step is to use the property of exponents that states for any nonzero real number \(a\) and any real number \(m\), $$ \frac{1}{a^m} = a^{-m} $$ Applying this property to our expression, we get $$ x^{-\frac{1}{2}} $$ So our equivalent expression in the form of \(x^n\) is $$ x^{-\frac{1}{2}} $$ and the value of \(n\) is \(-\frac{1}{2}\).

Key concepts

Properties of Exponents

Understanding the properties of exponents is like having a toolbox for handling powers and roots of numbers more effectively. One such important property is that of negative exponents. If you have the expression \( \frac{1}{a^m} \), it can be rewritten as \( a^{-m} \). This means that reciprocals of powers can be expressed as negative exponents.
Another crucial property is the rule for multiplying powers with the same base. If you have \( a^m \cdot a^n \), this can be simplified to \( a^{m+n} \). This property is not directly used in the original exercise but is handy when dealing with expressions involving multiple terms. By understanding these properties, you can effortlessly convert roots into fractional exponents, move terms around, and simplify complex expressions.
These properties help in algebra because they provide shortcuts and make expressions more manageable, especially in simplifying or expanding them.

Fractional Exponents

Fractional exponents might seem intimidating at first, but they are quite simple once you understand what they represent. A fractional exponent, such as \( x^{\frac{1}{2}} \), is just another way of writing a root. Specifically, \( x^{\frac{1}{2}} \) means the square root of \( x \).
This method of writing roots can be extended to other roots too. For instance, \( x^{\frac{1}{3}} \) is the cube root of \( x \), and \( x^{\frac{m}{n}} \) represents the \( n \)-th root of \( x^m \). Using fractional exponents can streamline calculations and simplify expressions because it's easier to manipulate exponents than roots in many cases.
In the exercise provided, converting \( \sqrt{x} \) into \( x^{\frac{1}{2}} \) makes it easier to apply further exponent rules, such as the property of negative exponents.

Simplifying Expressions

Simplifying an expression involves reducing it to its most basic form without changing its value. By using the properties of exponents discussed earlier, we can transform expressions into simpler, more workable forms.
In the given exercise, simplifying \( \frac{1}{\sqrt{x}} \) involves transforming it into an expression that is easier to handle, such as \( x^{-\frac{1}{2}} \). This simplification process begins by converting the square root into a fractional exponent \( x^{\frac{1}{2}} \).
Next, using the property of exponents, \( \frac{1}{a^m} = a^{-m} \), helps us to rewrite the expression as a single power of \( x \), making it more straightforward to understand and work with. The end result is an expression where everything is expressed as a power of \( x \), which is often a goal in algebra to make further operations, calculations, or comparisons more straightforward.