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Chapter 9: Problem 31

Simplify. $$ \sqrt{a^{14}} $$

Short Answer

Expert verified
The simplified form of \(\sqrt{a^{14}}\) is \(a^7\).

Step by step solution

01

Understand the square root

The square root of a number is a value that, when multiplied by itself, gives the original number. For any non-negative integer value, \(\sqrt{a} = a^{1/2}\).
02

Apply the exponent rule inside the square root

Recall the exponent rule for square roots: \(\sqrt{a^n} = a^{n/2}\). To apply this rule, set \(n = 14\) in the given expression \(\sqrt{a^{14}}\).
03

Simplify the exponent

Using the rule from Step 2, \(\sqrt{a^{14}} = a^{14/2}\).
04

Calculate the exponent result

Simplify the fraction: \(14/2 = 7\). Thus, \(a^{14/2} = a^7\).

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Key Concepts

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

Understanding Square Roots
Square roots are fundamental in simplifying expressions involving exponents. A square root of a number is another number that, when multiplied by itself, results in the original number. For example, \(\sqrt{9}=3\) because \(3\times3=9\). In algebra, square roots are represented as \(\sqrt{a}\), which can also be written as \(a^{1/2}\). This notation is particularly useful when dealing with exponents, allowing us to apply exponent rules more easily.
In the given exercise, we see \(\sqrt{a^{14}}\). We can simplify this by using the rule \(\sqrt{a^n} = a^{n/2}\). This way, we turn the square root operation into a simple exponent division.
Applying Exponent Rules
Exponent rules help us manage and simplify expressions involving powers. Here are a few fundamental rules to remember:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \( \left( a^m \right)^n = a^{mn} \)
  • Quotient of Powers: \(\frac{a^m}{a^n}=a^{m-n}\)
  • Power of a Product: \( (ab)^m = a^m b^m \)

In our problem, we use the square root rule for exponents: \( \sqrt{a^n} = a^{n/2} \). Applying this rule to \(\sqrt{a^{14}}\), we simplify it to \(a^{14/2} \). Finally, we calculate the exponent division: \(14/2 = 7\), giving us the simplified form \(a^7 \).
Remembering these rules makes handling exponents straightforward and aids in simplifying complex expressions.
Fraction Operations Simplified
When dealing with exponents and square roots, we often encounter fraction operations. Mastering these makes the process smoother. Here鈥檚 a summary of basic fraction operations to refresh your memory:
  • Addition/Subtraction: Convert to common denominators before adding or subtracting the numerators
  • Multiplication: Multiply the numerators together and the denominators together
  • Division: Multiply by the reciprocal of the divisor

In our problem, the fraction involved is straightforward: \( \frac{14}{2} \). We divide the numerator by the denominator to get \( 7 \). This simple operation is essential because our final answer relies on accurately simplifying this fraction.
Understanding and practicing these basic operations ensure precision and confidence while simplifying expressions involving exponents and square roots.

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