Math Problem Statement

Bentuk sederhana dari (3a^2b^{-3})^3 × (2a)^{-2} dalam bentuk eksponen positif adalah . . .

Solution

The problem in the image asks to simplify the expression (3a2b3)3×(2a)2(3a^2b^{-3})^3 \times (2a)^{-2} into a form with positive exponents. Let's break it down step by step:

Expression:

(3a2b3)3×(2a)2(3a^2b^{-3})^3 \times (2a)^{-2}

Step 1: Simplify each term

  • First term: (3a2b3)3(3a^2b^{-3})^3
    Apply the power rule to each factor inside the parentheses: (3a2b3)3=33(a2)3(b3)3=27a6b9(3a^2b^{-3})^3 = 3^3 \cdot (a^2)^3 \cdot (b^{-3})^3 = 27a^6b^{-9}

  • Second term: (2a)2(2a)^{-2}
    Apply the power to each factor: (2a)2=22a2=122a2=14a2(2a)^{-2} = 2^{-2} \cdot a^{-2} = \frac{1}{2^2 \cdot a^2} = \frac{1}{4a^2}

Step 2: Combine both terms

Now, multiply the results from both terms: 27a6b9×14a2=27a6b94a227a^6b^{-9} \times \frac{1}{4a^2} = \frac{27a^6b^{-9}}{4a^2}

Step 3: Simplify the exponents

Use the exponent rules to simplify the powers of aa: 27a64a2=274a62b9=274a4b9\frac{27a^6}{4a^2} = \frac{27}{4}a^{6-2}b^{-9} = \frac{27}{4}a^4b^{-9}

Step 4: Write the expression with positive exponents

Since the exponent of bb is negative, move it to the denominator: 27a44b9\frac{27a^4}{4b^9}

Thus, the simplified form is: 27a44b9\frac{27a^4}{4b^9}

Conclusion:

The correct option is C, which matches 27a44b9\frac{27a^4}{4b^9}.


Would you like further clarification on any step?

Here are five related questions to consider:

  1. What are the general rules for multiplying exponents?
  2. How do negative exponents affect a term's position in a fraction?
  3. What happens when you raise a product of terms to a power?
  4. Can you simplify (ambn)p(a^m \cdot b^n)^p?
  5. How do fractional exponents relate to roots?

Tip: Always apply the power rule individually to each factor within parentheses before simplifying.

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Math Problem Analysis

Mathematical Concepts

Exponents
Simplifying Expressions
Algebra

Formulas

Power of a Power Rule: (a^m)^n = a^{m*n}
Multiplying Powers with the Same Base: a^m * a^n = a^{m+n}
Negative Exponent Rule: a^{-n} = 1/a^n

Theorems

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Suitable Grade Level

Grades 8-10