Problem 110 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{p^{20}}{p^{10}}} $$

Step-by-Step Solution

Answer

\(\root 2 \relax {\frac{p^{20}}{p^{10}}} = p^5\).

Step 1: Simplify the Fraction Inside the Square Root

To begin, simplify the fraction \(\frac{p^{20}}{p^{10}}\). This can be done by subtracting the exponent in the denominator from the exponent in the numerator, using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\). Therefore: \(\frac{p^{20}}{p^{10}} = p^{20-10} = p^{10}\).

Step 2: Apply the Square Root

Next, take the square root of \(p^{10}\). Use the property \(\root n \relax {a^m} = a^{m/n}\) This gives: \(\root 2 \relax {p^{10}} = p^{10/2} = p^5\).

Key Concepts

Simplifying Exponents

Exponents are used to represent repeated multiplication. Simplifying them means making the expression easier to understand and work with. To simplify an exponent like \(\frac{p^{20}}{p^{10}}\), apply the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\). This combines the two exponents by subtracting the denominator from the numerator, resulting in \(\frac{p^{20}}{p^{10}} = p^{20-10} = p^{10}\). Simplifying exponents this way can make solving problems faster and easier.

Square Roots

A square root is a number that, when multiplied by itself, gives the original number. For example, \(\root 2 \relax {9} = 3\) because \((3 \times 3 = 9)\). In the context of exponents, taking the square root is similar to dividing the exponent by 2. For instance, \(\root 2 \relax {p^{10}} = p^{10/2} = p^5\). This uses the property \(\root n \relax {a^m} = a^{m/n}\). Simplifying square roots helps in making complex expressions more manageable and understandable. Bullet points make understanding key points easier:

• Square root of 16 is 4
• Square root of \(\frac{36}{4}\) is \(\frac{6}{2} = 3\)

Breaking down these concepts provides a clear pathway to understanding.

Properties of Exponents

Understanding the properties of exponents is crucial to simplifying and solving algebraic expressions. Here are some key properties:

• The product of powers property: \((a^m \times a^n = a^{m+n})\)
• The quotient of powers property: \(\frac{a^m}{a^n} = a^{m-n}\)
• The power of a power property: \((a^m)^n = a^{mn}\)

Using these properties can help transform complex algebraic expressions into simpler forms. For instance, in the given exercise, we used the quotient of powers property to simplify \(\frac{p^{20}}{p^{10}} = p^{10}\). Familiarity with these rules makes solving exponent-related problems straightforward.

Fractional Exponents

Fractional exponents are another way to write roots. For example, \(\root 2 \relax {x}\) is the same as \((x^{1/2})\). This notation can make it easier to apply rules for solving equations and expressions. For the exercise, we used the property \(\root n \relax {a^m} = a^{m/n}\) to find that \(\root 2 \relax {p^{10}}\) simplifies to \({p^{10/2}} = p^5\). When working with fractional exponents, remember:

• \