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Chapter 8: Problem 37

Simplify. $$\sqrt{a^{22}}$$

Short Answer

Expert verified
The simplified form is \(\text{a}^11\).

Step by step solution

01

Understanding the Square Root

The square root of a number can be defined as a number which produces the original number when it is squared. In this case, simplifying the square root means finding an equivalent expression which, when squared, gives the original expression inside the square root.
02

Applying the Square Root to the Exponent

Utilize the property of exponents that \(\frac{m}{n}\textrm{ power of a variable means that n-factor root of the variable power raised to m}\). Having \(\frac{22}{2}\textrm{ or 11 be the exponent}\). This results in the expression \(\text{simplified } a^{11}.\)
03

Simplify

After applying the exponent rule, the simplified form of \(\text{ such expression becomes } a^{11}\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is all about making them easier to read and work with. In algebra, we often have complex expressions that can look intimidating. But by following some rules and breaking them down step by step, we can simplify them into more manageable forms. For example, in the exercise where we simplify \(\text{\sqrt{a^{22}}}\), we are essentially trying to find an equivalent, simpler form of that expression.
Exponent Rules
Exponent rules are a set of mathematical laws that govern how to handle powers of numbers or variables. One important rule used in our exercise is the square root property. The property of exponents tells us that \(\text{\sqrt{x^m} = x^{m/2}}\).

In our example, we have \(\text{\sqrt{a^{22}}}\). Using the exponent rule, this expression can be rewritten as \(\text{a^{22/2}}\), which simplifies to \(\text{a^{11}}\).

Other common exponent rules you might come across include:
  • Multiplying powers with the same base: \(a^m \times a^n = a^{m+n}\).
  • Dividing powers with the same base: \(a^m / a^n = a^{m-n}\).
  • Power of a power: \( (a^m)^n = a^{mn}\).
Algebraic Writing
Algebraic writing is all about expressing mathematical ideas clearly and efficiently. When dealing with square roots and exponents, it's important to write your expressions in a way that follows standard mathematical conventions.

For instance, in the exercise where we simplified \(\text{\sqrt{a^{22}}}\), we used proper algebraic notation to show our work clearly. We started with the original expression, applied the exponent rule, and wrote the simplified form correctly.

Good algebraic writing includes:
  • Using proper symbols and notation.
  • Following the order of operations.
  • Clearly showing each step in your solution process.
Writing your algebraic solutions this way helps ensure that anyone reading your work can easily follow your thought process and understand how you arrived at your answer.

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