Math Problem Statement

Simplify the expression \( \frac{3a^{-\frac{2}{3}} + 2a^{\frac{1}{3}}}{a^{-\frac{5}{3}}} \).

Solution

The given expression is:

3a23+2a13a53\frac{3a^{-\frac{2}{3}} + 2a^{\frac{1}{3}}}{a^{-\frac{5}{3}}}

Let's break it down and simplify the expression step by step.

Step 1: Simplify the numerator

The numerator consists of two terms: 3a233a^{-\frac{2}{3}} and 2a132a^{\frac{1}{3}}. It remains as:

3a23+2a133a^{-\frac{2}{3}} + 2a^{\frac{1}{3}}

Step 2: Simplify the denominator

The denominator is a53a^{-\frac{5}{3}}.

Step 3: Apply the exponent rule

Use the property of exponents that states:

xmxn=xmn\frac{x^m}{x^n} = x^{m-n}

We can rewrite the whole fraction as:

(3a23+2a13a53)=(3a23+2a13)a53\left( \frac{3a^{-\frac{2}{3}} + 2a^{\frac{1}{3}}}{a^{-\frac{5}{3}}} \right) = (3a^{-\frac{2}{3}} + 2a^{\frac{1}{3}}) \cdot a^{\frac{5}{3}}

Step 4: Distribute the a53a^{\frac{5}{3}}

We distribute a53a^{\frac{5}{3}} to each term in the numerator:

  1. 3a23a53=3a(23+53)=3a33=3a1=3a3a^{-\frac{2}{3}} \cdot a^{\frac{5}{3}} = 3a^{\left(-\frac{2}{3} + \frac{5}{3}\right)} = 3a^{\frac{3}{3}} = 3a^1 = 3a
  2. 2a13a53=2a(13+53)=2a63=2a22a^{\frac{1}{3}} \cdot a^{\frac{5}{3}} = 2a^{\left(\frac{1}{3} + \frac{5}{3}\right)} = 2a^{\frac{6}{3}} = 2a^2

Step 5: Final Simplified Expression

Now, combine the simplified terms:

3a+2a23a + 2a^2

Thus, the final simplified expression is:

2a2+3a2a^2 + 3a

Would you like further details or clarifications on any step?

Here are 5 related questions you might consider:

  1. How do exponent rules work for fractional exponents?
  2. How can we handle negative exponents in more complex expressions?
  3. What happens when you multiply terms with different bases and exponents?
  4. Can we factor the final expression 2a2+3a2a^2 + 3a?
  5. How can this technique be applied to more advanced algebraic simplifications?

Tip: Always simplify terms involving exponents first, as this can help avoid unnecessary complexity in the solution.

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Math Problem Analysis

Mathematical Concepts

Exponents
Algebraic Simplification
Fractional Exponents

Formulas

Exponent rule: \( \frac{x^m}{x^n} = x^{m-n} \)
Distributive property of multiplication over addition

Theorems

Properties of exponents

Suitable Grade Level

Grades 10-12