Problem 68 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{a^{7} b^{6}}}{\sqrt{a^{3} b^{2}}}\)

Step-by-Step Solution

Answer

The simplified expression is \( a^2 b^2 \).

Step 1: Apply the Quotient Rule for Square Roots

The quotient rule for square roots states \( \sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}} \). Therefore, apply this rule to separate the square roots in the numerator and denominator: \( \frac{\sqrt{a^{7} b^{6}}}{\sqrt{a^{3} b^{2}}} = \sqrt{\frac{a^{7} b^{6}}{a^{3} b^{2}}} \).

Step 2: Simplify the Expression Inside the Square Root

Inside the square root, simplify the expression by dividing the powers of each variable: \( \frac{a^7 b^6}{a^3 b^2} = a^{7-3} b^{6-2} = a^4 b^4 \).

Step 3: Calculate the Square Root of the Simplified Expression

Now take the square root of each part of the expression: \( \sqrt{a^4 b^4} = \sqrt{a^4} \times \sqrt{b^4} \). This simplifies to \( a^{4/2} b^{4/2} = a^2 b^2 \).

Step 4: Verify and Present the Simplified Answer

Ensure that the expression \( a^2 b^2 \) is the simplest form possible. There are no remaining parts to simplify further.

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions is all about making them easier to work with, without changing their value. In the context of dividing square roots, you essentially condense a complex fraction or equation into a simpler, more digestible form.

Start by applying the Quotient Rule for square roots, such as \( \sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}} \). This simplifies calculations by allowing you to directly work with more straightforward parts of the expression, one at a time.

Once you've applied this rule, you'll often need to simplify by canceling out terms or combining like terms. In our example, after separating the square roots, you'd simplify further by dividing the exponents according to the rules of exponents, which leads us to the next crucial concept.

Exponents

Exponents are a shorthand way to denote repeated multiplication. For example, the expression \( a^4 \) means \( a \times a \times a \times a \). They are governed by a set of rules that apply when you multiply, divide, or raise powers.

• When dividing terms with the same base, deduct the exponent in the denominator from the exponent in the numerator, \( a^m / a^n = a^{m-n} \).
• To extract roots, divide the exponent by the root's degree. For a square root, you'd halve the exponent, \( \sqrt{a^6} = a^{3} \).

These rules make calculations faster and can save a lot of time and effort when simplifying expressions involving exponents in algebraic fractions.

Square Roots

Square roots return a value that, when multiplied by itself, gives the original number. They are represented by the radical symbol (\( \sqrt{} \)), and dealing with them involves special rules, especially in algebra.

A key rule when working with square roots is simplifying the expression inside before applying the root. This may involve using the products and quotients of roots. For instance, \( \sqrt{a^4} \) simplifies to \( a^{4/2} = a^2 \). By simplifying inside the root first, you apply the root more effectively.

Always remember: removing a square root entirely often involves rewriting it in exponential form and then simplifying, as we did in the exercise by moving from \( \sqrt{a^4 b^4} \) to \( a^2 b^2 \). Recognizing these patterns aids in mastering algebraic manipulations involving roots.