Problem 8 - Elementary Algebra

Question

Simplify the square root expressions. $$ \frac{\sqrt{a^{3 m-5}}}{\sqrt{a^{m-1}}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{\sqrt{a^{3m-5}}}{\sqrt{a^{m-1}}}\). Answer: \(\sqrt{a^{2m-4}}\)

Step 1: Combine the square root expressions

We can combine the square roots into one expression using the following property: \(\sqrt{a}/\sqrt{b}=\sqrt{a/b}\). Apply this property to the given expression: $$ \frac{\sqrt{a^{3 m-5}}}{\sqrt{a^{m-1}}}=\sqrt{\frac{a^{3m-5}}{a^{m-1}}} $$

Step 2: Simplify the expression under the square root using the properties of exponents

For a given base, use the following property of exponents: \(\frac{a^{n}}{a^{m}}=a^{n-m}\). Apply this property to the expression under the square root: $$ \sqrt{\frac{a^{3m-5}}{a^{m-1}}}=\sqrt{a^{(3m-5)-(m-1)}} $$

Step 3: Simplify the exponent

Carry out the subtraction in the exponent: $$ \sqrt{a^{(3m-5)-(m-1)}}=\sqrt{a^{3m-5-m+1}} $$ Combine the like terms: $$ \sqrt{a^{3m-5-m+1}}=\sqrt{a^{2m-4}} $$

Step 4: Final answer

The simplified expression is: $$ \frac{\sqrt{a^{3m-5}}}{\sqrt{a^{m-1}}}=\sqrt{a^{2m-4}} $$

Key Concepts

Combining Square Roots

When working with square roots in algebra, there are several processes that can help simplify complex expressions.
One key technique involves combining multiple square roots into a single square root expression.

In the given problem, we have the expression \(\frac{\sqrt{a^{3m-5}}}{\sqrt{a^{m-1}}}\). By using the property \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), we can combine the two separate square roots into one:

• This step simplifies the division of the individual roots into a single root of a quotient.
• It lays the groundwork for further simplification of the expression inside the square root.

Combining square roots is helpful because it allows us to manage and reduce the complexity of an expression at an early stage in the simplification process.

Properties of Exponents

Understanding the properties of exponents is crucial when simplifying expressions that involve powers.
The fundamental rule used in this exercise is the quotient of powers property, which states that \(\frac{a^{n}}{a^{m}}=a^{n-m}\).

Let's apply this to the expression under the square root:
\[\sqrt{\frac{a^{3m-5}}{a^{m-1}}} = \sqrt{a^{(3m-5) - (m-1)}}\]

• This reduction happens because the same base \(a\) is present in both the numerator and denominator.
• The exponents undergo subtraction, thus simplifying the power inside the square root.

By organizing exponents in this manner, we can make a more streamlined expression that is closer to the final simplified form.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form.
After applying the properties of exponents, it is important to carry out any outstanding arithmetic to finalize the simplification.

In the exercise, the expression \(\sqrt{a^{(3m-5)-(m-1)}}\) was transformed to \(\sqrt{a^{2m-4}}\) by simplifying the exponent:

• First, we did any addition or subtraction within the exponent, in this case: \((3m-5) - (m-1) = 3m - 5 - m + 1\).
• This calculation led to \(2m - 4\), which then became the single exponent in the root \(\sqrt{a^{2m-4}}\).

This process is crucial because it ensures that the expression is as simple as possible.
It eliminates potential errors and prepares the equation for further algebraic operations if necessary.