Problem 82 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{\frac{72 q^{7}}{25 q^{3}}} $$

Step-by-Step Solution

Answer

\(\frac{6q^2\sqrt{2}}{5}\)

Step 1: Simplify the Radical Expression Inside

Replace the division within the square root by reducing the fractions inside. Begin by simplifying the expression under the square root: \[ \frac{72q^7}{25q^3} = \frac{72}{25} \cdot \frac{q^7}{q^3} \]Simplify the variable part by subtracting the exponents: \[ q^7 / q^3 = q^{7-3} = q^4 \]So the expression becomes \[ \frac{72q^4}{25} \]

Step 2: Separate the Square Root Terms

Next, separate the numbers and variable parts under the square root sign. \[ \sqrt{\frac{72q^4}{25}} = \sqrt{\frac{72}{25}} \times \sqrt{q^4} \]

Step 3: Solve the Numerical Square Root

Simplify the square root of the fraction involving numbers. First, simplify \(\sqrt{72}\) by recognizing it involves the square root of known perfect squares: \[ 72 = 36 \times 2 \quad \Rightarrow \quad \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \]Now, simplify the denominator:\[ \sqrt{25} = 5 \]Put these together:\[ \sqrt{\frac{72}{25}} = \frac{6\sqrt{2}}{5} \]

Step 4: Solve the Variable Term's Square Root

Now simplify the \(\sqrt{q^4}\) term by recognizing that \(\sqrt{q^4} = q^{4/2} = q^2\). This yields: \[ \sqrt{q^4} = q^2 \]

Step 5: Combine Everything

Finally, combine the simplified results of steps 3 and 4.\[ \sqrt{\frac{72q^7}{25q^3}} = \frac{6\sqrt{2}}{5} \times q^2 = \frac{6q^2\sqrt{2}}{5} \]Thus, the simplified expression is: \[ \frac{6q^2\sqrt{2}}{5} \]

Key Concepts

Understanding Radical Expressions

Radical expressions might seem intimidating at first, but once you grasp the basic idea, they become much easier to handle. A radical expression involves roots, such as square roots or cube roots. The most common is the square root, denoted by the symbol \(\sqrt{ }\). It signifies that we are looking for a number which, when multiplied by itself, gives the original number under the root.

When dealing with radical expressions:

• Focus on the numbers first. Break down numbers under the square root to their prime factors to identify perfect squares or other root values.
• Remember to handle each part of the expression separately, especially when variables and coefficients are involved. Simplify expressions step by step to avoid errors.
• For expressions involving fractions under a square root, treat the numerator and denominator as separate radical expressions.

When you break down a radical expression like \(\sqrt{\frac{72q^4}{25}}\), you separate the number and the variables to simplify each part individually. The aim is to make the expression as simple as possible.

Mastering Exponent Rules

Exponents tell us how many times to multiply a number by itself. Understanding exponent rules is crucial when working with expressions involving powers and roots.

Some key exponent rules include:

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

In our expression, \(q^7/q^3\), the quotient of powers rule is used, which allows us to subtract the exponents: \(q^{7-3} = q^4\). This simplification forms a crucial step in working through the rational expression and finding its simplest form.

Applying these rules accurately is the key to solving complex algebraic problems.

Simplifying with Square Roots

The square root is one of the most common types of roots. It is essential to know how to simplify expressions that involve square roots, as many algebraic equations will require this skill.

When simplifying square roots:

• Break down numbers into their prime factors and identify the perfect squares that can be simplified. For example, \(\sqrt{72}\) can be simplified by recognizing \(72 = 36 \times 2 = 6^2 \times 2\), which simplifies the square root to \(6\sqrt{2}\).
• Simplify expressions with variables by applying the rule that \(\sqrt{x^n} = x^{n/2}\). For instance, \(\sqrt{q^4}\) becomes \(q^{2}\) because \(4 \div 2 = 2\).
• If an expression includes a fraction, address the numerator and denominator separately, then combine them back after simplification, as shown when transforming \(\sqrt{\frac{72}{25}}\) into \(\frac{6\sqrt{2}}{5}\).

By mastering these strategies, you'll be able to simplify any square root expression efficiently and correctly.