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Chapter 8: Problem 39

Simplify. $$\sqrt{a^{23}}$$

Short Answer

Expert verified
The simplified form is \( a^{23/2} \).

Step by step solution

01

Understand the given expression

The expression given is \(\[\begin{equation} \sqrt{a^{23}} \end{equation}\] \). This is a square root of a variable raised to a power.
02

Apply the square root property

Recall that \(\[\begin{equation} \sqrt{x^{n}} = x^{n/2} \end{equation}\] \). This property helps in simplifying expressions under a square root.
03

Simplify the exponent

Using the property in Step 2, substitute the value of \( x \) with \( a \) and \( n \) with \( 23 \). Thus, we get: \(\[\begin{equation} \sqrt{a^{23}} = a^{23/2} \end{equation}\] \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Introduction to Square Roots
Square roots are a fundamental concept in mathematics, particularly in algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. In algebraic expressions, the square root is often represented by the radical symbol (鈭�). For instance, the square root of 16 is written as \(\sqrt{16} = 4\). When dealing with variables and exponents, the property \[ \sqrt{x^n} = x^{n/2} \] is key. This means that taking the square root of a variable to any power simply involves dividing the exponent by 2. By understanding this rule, you can simplify complex expressions easily.
Understanding Exponent Rules
Exponent rules are essential tools for manipulating expressions involving powers. There are several key rules to remember:
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Power of a Power: \( (a^m)^n = a^{m路n}\)
  • Power of a Product: \( (ab)^n = a^n \cdot b^n\)
In our problem, we use the property \[ \sqrt{x^n} = x^{n/2} \]. Applying this to the expression \(\sqrt{a^{23}}\), we get \(a^{23/2}\), which simplifies the expression significantly. Such conversions are a big part of working with exponents in algebra.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a fundamental skill in algebra. It involves reducing expressions to their simplest form to make them easier to work with. This often means applying exponent rules, combining like terms, and factoring. Let's break down the process using our example: \(\sqrt{a^{23}}\):
  • First, identify the given expression and understand the operation involved (square root here).
  • Next, recall the relevant property: \[ \sqrt{x^n} = x^{n/2} \].
  • Apply this property to our expression: \(\sqrt{a^{23}} = a^{23/2}\).
  • The result is \(a^{23/2}\), which is the simplified form.
By following these steps, you can simplify almost any algebraic expression involving exponents and roots.

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