Problem 115 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{96 p^{9}}{6 p}} $$

Step-by-Step Solution

Answer

The simplified expression is \(4 p^4\).

Step 1: Simplify the fraction inside the square root

First, simplify the fraction \(\frac{96 p^9}{6 p}\). Divide the coefficients and subtract the exponents of the same bases. \(\frac{96}{6} = 16\) and \(\frac{p^9}{p} = p^{8}\). The expression becomes \(\frac{96 p^9}{6 p} = 16 p^8\).

Step 2: Apply the square root

Next, apply the square root to the simplified expression \(\frac{96 p^9}{6 p} = 16 p^8\). This means we need to find the square root of each term separately: \(\sqrt{16 p^8}\).

Step 3: Simplify the square root

Simplify \(\sqrt{16 p^8}\). The square root of 16 is 4 because \(\sqrt{16} = 4\), and the square root of \(\ p^8 \) is \(\ p^4 \) because \(\sqrt{p^8} = p^4\). Therefore, \(\sqrt{16 p^8} = 4 p^4\).

Key Concepts

Square Roots

Square roots are numbers that produce a specified quantity when multiplied by themselves. They are denoted using the radical symbol (\sqrt{}). For any positive number \(a\), the principal square root is \(\sqrt{a}\) which means finding a number \(b\) such that \(b^2 = a\).
When applying square roots to expressions, it's necessary to handle each component individually. For instance, the square root of a product is the product of the square roots. For \(\sqrt{16 p^8}\), the square root of 16 is 4 because 4 \times 4 = 16.
Similarly, for \(p^8\), since \(p^8\) is equivalent to \(p^4 \times p^4\), the square root is \(p^4\). Combining these results gives us \(\sqrt{16 p^8} = 4 p^4\).

Fraction Simplification

Fraction simplification involves reducing fractions to their simplest form. Start by simplifying each part of the fraction. Take \(\frac{96 p^9}{6 p}\); both numerator and denominator share common factors. Divide 96 by 6 to get 16. Next, simplify \(p^9 \div p\). Using exponent rules (subtracting exponents with the same base), we get \(p^{9-1} = p^8\).
The fraction then simplifies to \(\frac{16 p^8}{1} = 16 p^8\). This earlier simplification makes applying further operations like square roots straightforward and quicker.

Exponent Rules

Exponent rules are essential for simplifying expressions with powers. Here's a quick overview of some essential rules:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \( (a^m)^n = a^{m \times n}\)

In this problem, \(\frac{p^9}{p} = p^{9-1} = p^8\), exemplifies the quotient of powers rule. This enables simplifying complex fractional exponents effortlessly. By understanding and applying these rules, students can tackle sophisticated algebraic expressions with confidence.