Problem 52 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ \left(3 a^{3 / 4}\right)^{2}\left(5 a^{1 / 2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(45a^2\).

Step 1: Expand Exponents

Begin by expanding the exponents inside the expression. The first part of the expression \((3a^{3/4})^2\) can be expanded using the power rule \((a^m)^n = a^{m\times n}\),giving \(3^2 \cdot a^{2\times(3/4)} = 9a^{3/2}\). The second part, \(5a^{1/2}\) remains the same for now.

Step 2: Combine Like Terms

Now, combine the like terms obtained from the expanded expression. Multiply the constant terms and add the powers of the like bases using the rule: \(a^m \cdot a^n = a^{m+n}\). The expression becomes \(9 \cdot 5 \cdot a^{3/2 + 1/2} = 45a^{2}\).

Step 3: Simplify the Expression

Verify that the expression is fully simplified. The exponents are now positive, and integer exponents replace any fractional exponents during exponent addition. The final simplified expression is \(45a^2\).

Key Concepts

Power Rule

The Power Rule is a foundation of working with exponents. It states that when you raise a power by another power, you multiply the exponents. This is expressed mathematically as \[ (a^m)^n = a^{m\times n} \].

For example, in the problem, we apply the Power Rule to \((3a^{3/4})^2\).

• First, keep the coefficient as it is, but raise it to the power if applicable. Here, 3 is raised to the power of 2, resulting in 9.
• For the variable part, multiply the exponent \(\frac{3}{4}\) by the power 2. This gives \(a^{\frac{3}{2}}\), simplifying the expression to \(9a^{3/2}\).

Understanding the Power Rule makes simplifying more complex expressions manageable and helps ensure no steps are missed during simplification.

Combining Like Terms

Once you have expanded the expression using the Power Rule, combining like terms is the next step. The goal is to simplify terms with the same bases by combining them using addition of their exponents.

• Use the rule \( a^m \cdot a^n = a^{m+n} \) to add the exponents of like bases.
• Take the expression \(9a^{3/2}\cdot 5a^{1/2}\).
• Combine coefficients: \(9\cdot5 = 45\).
• Add the exponents of \(a\): \( \frac{3}{2} + \frac{1}{2} = 2\). This results in \(45a^2\).

By extracting and adding like terms, you create a simpler expression that is easier to work with and find solutions.

Simplifying Expressions

Simplifying expressions involves bringing an expression to its simplest form, where all calculations are completed, and the result has the fewest possible terms and simplest form of exponents. This process often involves several steps:

• Make sure all exponents are positive and simplified.
• Ensure the coefficients are multiplied accurately.
• Simplify the variable parts by combining terms and using the rules for exponents.

In our problem, we started with multiple components and operated on them to simplify down to \(45a^2\). The expression doesn't just get simpler; it also becomes more effective for use in solving equations or further manipulations.
It's important to revisit each step to ensure the expression is fully simplified, catching any negative exponents or unsimplified fractions, so the expression is always presented in its most efficient form.