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Chapter 8: Problem 35

Simplify. $$\sqrt{a^{21}}$$

Short Answer

Expert verified
\a^{10.5}\

Step by step solution

01

Identify the Expression

The given expression is \(\root{\ a^{21}}\).
02

Rewrite Under the Radical

Rewrite the expression under the square root as \( (\root{a^{21}} = {\root{(a^{10.5} ) \cdot \ (a^{10.5})}})\)
03

Simplify

Since the square root of \( (a^{10.5} ) \cdot \ (a^{10.5})\) can be simplified to \a^{10.5}\.

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Key Concepts

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

Square Root
The square root is a number that produces a specified quantity when multiplied by itself. For example, the square root of 9 is 3 because 3 * 3 = 9. A square root is represented by the symbol \(\root{}\rupees{}\).
The square root of a variable with an exponent can be simplified using the properties of exponents. In our exercise, we start with \(\root{a^{21}}\), meaning we want to find a value that, when squared, results in \(a^{21}\).
This requires understanding that \(\root{x^2} = x\) when x is positive. With this concept, you can work towards simplifying radicals more effectively. The key when using square roots is to recognize pairs of terms and their patterns.
Exponents
An exponent indicates how many times to multiply a number by itself. For example, \(a^2\) means \(a \times a\). When dealing with exponents under a square root, there are several rules that can help simplify the expressions.
Here are some useful exponent rules:
  • \(a^m a^n = a^{(m+n)}\)
  • \( (a^m)^n = a^{(mn)}\)
  • \({a^m} = a^{(m/n)}\) when under a root sign
Applying these rules to our exercise, we have \(\root{a^{21}}\). We rewrite the expression inside as \((a^{10.5} \times a^{10.5})\). This uses the fact that \(a^{21}\) equals \(a^{10.5 \times 2}\).
At this point, it's simply a matter of recognizing that \(\root{a^{(10.5 \times 2)}} = a^{10.5}\). Understanding these exponent rules will give you a strong foundation in algebraic simplification.
Algebraic Simplification
Algebraic simplification involves reducing expressions to their simplest form. This often includes combining like terms, factoring, or using various algebraic rules and properties.
In the given exercise, we applied a simplified path to make \(\root{a^{21}}\) more manageable. Here are the steps broken down:
  • Write \(\root{a^{21}}\) as \(a^{21/2}\)
  • Recognize that 21 divided by 2 gives us \(10.5\)
  • Thus, \(\root{a^{21}} = a^{10.5}\) because square rooting and squaring are inverse operations.
Remember this when you're given more complex problems. Always look for ways to break down the expression using the rules of exponents and square roots to simplify it step by step.
Practicing these simplifications will make handling algebraic expressions and radicals feel less daunting.

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