Problem 112 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{q^{8}}{q^{14}}} $$

Step-by-Step Solution

Answer

\( \frac{1}{q^{3}} \)

Step 1: Identify the Expression Inside the Square Root

The given expression is \( \frac{q^{8}}{q^{14}} \) inside the square root.

Step 2: Simplify the Fraction

Use the properties of exponents to simplify \( \frac{q^{8}}{q^{14}} \rightarrow q^{8-14} = q^{-6} \).

Step 3: Apply the Square Root

Now apply the square root: \( \sqrt{q^{-6}} \). This equals \( q^{-\frac{6}{2}} = q^{-3} \).

Step 4: Express with Positive Exponents

Finally, rewrite \( q^{-3} \) as \( \frac{1}{q^{3}} \) to express the answer with positive exponents.

Key Concepts

Properties of Exponents

When simplifying expressions involving exponents, understanding the properties of exponents is crucial. The main rules to remember are:

• Product of Powers: When multiplying like bases, add the exponents: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\)
• Power of a Product: To distribute the exponent over a product: \((ab)^m = a^m b^m\)
• Power of a Quotient: To distribute the exponent over a quotient: \((\frac{a}{b})^m = \frac{a^m}{b^m}\)

By applying these rules, you can simplify complex expressions like the one in the original exercise.

Simplification

Simplification is the process of making an expression easier to work with. Let's break down how we simplified the given expression step by step:
Step 1: Identify the fraction inside the square root: \sqrt{\frac{q^{8}}{q^{14}}}\
Step 2: Use the Quotient of Powers property. Subtract the exponents of like bases: \frac{q^{8}}{q^{14}} = q^{8-14} = q^{-6}\
This step reduces the fraction to a single term: \(q^{-6}\).
Step 3: Apply the square root to the simplified expression: \sqrt{q^{-6}}\. To find the square root, divide the exponent by 2: \sqrt{q^{-6}} = q^{-3}\.
Step 4: Finally, express the answer with a positive exponent. Recall: \(q^{-n} = \frac{1}{q^n}\). So, \(q^{-3} = \frac{1}{q^3}\).
After these steps, the expression is simplified to \frac{1}{q^3}\.

Positive Exponents

Positive exponents make expressions easier to interpret and work with. Here’s how to convert negative exponents to positive ones:

• Negative Exponents: A negative exponent indicates reciprocal: \(a^{-n} = \frac{1}{a^n}\)
• Rewriting Negatives: To rewrite \(q^{-3} \), convert it to its reciprocal: \(q^{-3} = \frac{1}{q^3}\)
• Consistency: Use positive exponents for final answers to maintain consistency and avoid confusion.

Converting to positive exponents is a crucial step, especially in algebra, to ensure clarity and correctness in mathematical expressions.