Problem 68 - Intermediate Algebra

Question

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt{a^{7} b^{6}}}{\sqrt{a^{3} b^{2}}} $$

Step-by-Step Solution

Answer

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

Step 1: Apply the Quotient Rule

To divide the square roots, we apply the quotient rule for square roots, which states that \( \frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}} \). So, we have: \[ \frac{\sqrt{a^7 b^6}}{\sqrt{a^3 b^2}} = \sqrt{\frac{a^7 b^6}{a^3 b^2}} \]

Step 2: Simplify the Exponents

We need to apply the division rule for exponents when the bases are the same, which is \( \frac{a^m}{a^n} = a^{m-n} \). Thus, we simplify:\[ \frac{a^7}{a^3} = a^{7-3} = a^4 \] and \[ \frac{b^6}{b^2} = b^{6-2} = b^4 \]So, our expression becomes:\[ \sqrt{a^4 b^4} \]

Step 3: Simplify the Square Root

At this step, we can simplify the square root. Recall that \( \sqrt{x^2} = |x| \), but since we are often dealing with positive results, we can simply take:\[ \sqrt{a^4 b^4} = \sqrt{(a^2 b^2)^2} = a^2 b^2 \]Therefore, the simplified expression is:\[ a^2 b^2 \]

Key Concepts

Simplifying Exponents

When simplifying exponents, a key idea is understanding the representation of powers. If we have a base raised to a certain power, like \(a^7\), it means we multiply \(a\) by itself 7 times. This foundational idea is essential when dealing with problems requiring simplification.

Simplifying exponents involves combining them to shorten expressions while maintaining their meaning. To simplify \(a^7\) divided by \(a^3\), we utilize the rule that says to subtract the exponents, as long as the bases are the same. Here's how it works:

• Divide by subtracting the exponents: \( \frac{a^7}{a^3} = a^{7-3} = a^4 \).
• For \(b^6\) divided by \(b^2\), perform the operation in the same manner: \( \frac{b^6}{b^2} = b^{6-2} = b^4 \).

Once both terms are simplified, they can be written together within the next operation, allowing us to simplify the entire expression.

Division Rule for Exponents

The division rule for exponents plays a crucial role when the problem involves dividing variables with the same base. This rule simplifies expressions in a neat way and reduces the chances of error.

The principle behind this rule is simple and effective: if you divide two exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator. In a general form, it states:

• \( \frac{x^m}{x^n} = x^{m-n} \).

This rule only applies when the bases are identical. You must ensure the bases are the same before applying this rule. This simplification is especially useful when you're dealing with big exponents, as it reduces large numbers into manageable expressions.

Simplifying Square Roots

Simplifying square roots is the final step after simplifying and dividing exponents. It's all about reducing the square roots to a simpler form.

To traverse this process, remember that the square root of a square is the absolute value of its base. For example, \( \sqrt{x^2} = |x| \). However, when simplifying math problems, we often work under the assumption that we're dealing with positive quantities, which allows us to discard the absolute value for ease:

• If you have \(\sqrt{a^4}\), since \((a^2)^2 = a^4\), it simplifies to \(a^2\).
• Likewise, \(\sqrt{b^4}\) simplifies to \(b^2\).

Thus, the expression \( \sqrt{a^4b^4} \) becomes \(a^2b^2\), making it much easier to handle and use in future calculations. Understanding these simplifications can greatly aid in solving and comprehending more complex problems.