Problem 69 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\frac{\sqrt{m n^{3}}}{a^{2} \sqrt{c^{-3}}} \cdot \frac{a^{-7} n^{-2}}{\sqrt{m^{2} c^{4}}}\)

Step-by-Step Solution

Answer

The simplified expression is: \( m^{-1/2} n^{-1/2} a^{-9} c^{-1/2} \)

Step 1: Simplify the Square Roots

Simplify the square roots in the expression \(\frac{\sqrt{m n^{3}}}{a^{2} \sqrt{c^{-3}}} \cdot \frac{a^{-7} n^{-2}}{\sqrt{m^{2} c^{4}}}\).- For \(\sqrt{m n^3}\), rewrite as \(\sqrt{m} \cdot \sqrt{n^3} = \sqrt{m} \cdot n^{3/2}\).- For \(\sqrt{c^{-3}}\), rewrite as \(c^{-3/2}\).- For \(\sqrt{m^2 c^4}\), rewrite as \(m \cdot c^2\).

Step 2: Substitute Simplified Square Roots

Substitute the simplified forms back into the expression:\[\frac{\sqrt{m} \cdot n^{3/2}}{a^2 \cdot c^{-3/2}} \cdot \frac{a^{-7} \cdot n^{-2}}{m \cdot c^2}\]

Step 3: Simplify Algebraic Fractions

Simplify by multiplying the fractions:- Combine terms:\(\frac{\sqrt{m} \cdot n^{3/2} \cdot a^{-7} \cdot n^{-2}}{a^2 \cdot c^{-3/2} \cdot m \cdot c^2}\)- Simplify further:\(\frac{\sqrt{m} \cdot n^{3/2 - 2} \cdot a^{-7}}{a^2 \cdot m \cdot c^{-3/2 + 2}}\)\(= \frac{\sqrt{m} \cdot n^{-1/2} \cdot a^{-9}}{m \cdot c^{1/2}}\)

Step 4: Simplify Radicals and Exponents

Now simplify:- \(\sqrt{m} = m^{1/2}\), thus:\[\frac{m^{1/2} \cdot n^{-1/2} \cdot a^{-9}}{m \cdot c^{1/2}}\]- Combine the \(m\) terms:\(m^{1/2}/m = m^{-1/2}\).- Final expression is:\[m^{-1/2} \cdot n^{-1/2} \cdot a^{-9} \cdot c^{-1/2}\]

Step 5: Final Solution

The completely simplified expression of the given problem is:\[ m^{-1/2} \cdot n^{-1/2} \cdot a^{-9} \cdot c^{-1/2} \]

Key Concepts

Radical Expressions

When dealing with radical expressions, we often encounter square roots or cube roots and so on. The radical symbol \( \sqrt{} \) denotes the square root operation, and we must be careful to simplify these expressions correctly. For example, the square root \( \sqrt{m n^3} \) can be broken down into \( \sqrt{m} \cdot \sqrt{n^3} \). Breaking down radicals is like breaking a large problem into smaller pieces - it helps us handle them better. For \( \sqrt{n^3} \), we use the rule \( \sqrt{n^3} = n^{3/2} \), simplifying this fractional exponent will allow us to further manage the expression.

It’s important to also note how negative exponents work with radicals. In the problem provided, \( \sqrt{c^{-3}} \) is reinterpreted using rules of radicals and exponents to become \( c^{-3/2} \). Similarly, \( \sqrt{m^2 c^4} \) can be simplified because \( \sqrt{m^2} = m \) and \( \sqrt{c^4} = c^2 \). By clearing radicals, we can more straightforwardly tackle the algebra that follows.

Exponent Rules

Exponent rules are essential for simplifying expressions. They allow us to manage powers and roots systematically. Some key rules include:

• \( a^m \cdot a^n = a^{m+n} \): when we multiply terms with the same base, we add their exponents.
• \( a^{m/n} = \sqrt[n]{a^m} \), which connects exponents to radicals.
• \( a^{-n} = \frac{1}{a^n} \), helping us rewrite negative exponents.

Understanding how these rules integrate into simplification is crucial. For example, in the problem, \( a^2 \cdot a^{-7} \) simplifies using \( a^{2-7} = a^{-5} \). Another great demonstration is in \( n^{3/2-2} \), which simplifies to \( n^{-1/2} \). Exponent combinations invite us to see expressions' hidden simplicity by transitioning them back to radical form where necessary, such as converting \( n^{-1/2} \) back to \( \frac{1}{\sqrt{n}} \). These efficiencies streamline our problem-solving process.

Fraction Simplification

Simplifying fractions, especially when they include algebraic expressions, involves both numerical and algebraic manipulation. The main idea is to reduce the fraction to its most simple form. This could involve:

• Canceling common terms in the numerator and denominator.
• Reducing negative exponents.

In our original expression, simplifying began by identifying overlapping terms. After substituting simplified radicals, you multiply numerators and denominators across, then step back to observe any terms that can be reduced. For example, cancel \( m^{1/2} \) over \( m \), resulting in \( m^{-1/2} \). Similar operations help reduce \( a \) and \( c \) terms via normal exponent simplification, resulting in \( a^{-9} \) and \( c^{-1/2} \) as seen in \( \frac{a^{-7} \cdot n^{-2}}{m \cdot c^2} \). Fraction simplification often nearly reduces problems to `bare bones,` and double-checking for errors at each step is key.