Math Problem Statement

Bentuk sederhana dari \left( \frac{9a^2 b^{-1} c}{27a^3 b^2 c^2} \right)^{-1} adalah....

Solution

The given mathematical expression to simplify is:

(9a2b1c27a3b2c2)1\left( \frac{9a^2 b^{-1} c}{27a^3 b^2 c^2} \right)^{-1}

Step 1: Simplify the fraction inside the parentheses

We begin by simplifying the fraction:

9a2b1c27a3b2c2\frac{9a^2 b^{-1} c}{27a^3 b^2 c^2}

This can be broken down as follows:

  • For the constants: 927=13\frac{9}{27} = \frac{1}{3}
  • For the powers of aa: a2a3=a23=a1\frac{a^2}{a^3} = a^{2-3} = a^{-1}
  • For the powers of bb: b1b2=b12=b3\frac{b^{-1}}{b^2} = b^{-1-2} = b^{-3}
  • For the powers of cc: cc2=c12=c1\frac{c}{c^2} = c^{1-2} = c^{-1}

Thus, the fraction simplifies to:

13a1b3c1\frac{1}{3} a^{-1} b^{-3} c^{-1}

Step 2: Apply the negative exponent

The expression now becomes:

(13a1b3c1)1\left( \frac{1}{3} a^{-1} b^{-3} c^{-1} \right)^{-1}

Using the property of exponents, (x1)=1x(x^{-1}) = \frac{1}{x}, the expression becomes:

3a1b3c1=3ab3c3 a^1 b^3 c^1 = 3ab^3c

Final Answer:

The simplified form is 3ab3c3ab^3c, which matches option A.

Let me know if you need more details or further clarification!


Here are 5 related questions for further practice:

  1. How do you simplify expressions with negative exponents?
  2. What is the rule for dividing terms with the same base but different exponents?
  3. How does multiplying by the reciprocal work when dealing with negative exponents?
  4. Can you simplify 4x22x3\frac{4x^{-2}}{2x^3}?
  5. What happens when you apply a power of negative one to a fraction?

Tip: Always combine like terms (same base) by adding or subtracting exponents when multiplying or dividing powers.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplifying Fractions
Exponents

Formulas

Exponent rules: a^m / a^n = a^(m-n), (x^-1) = 1/x
Fraction simplification

Theorems

Properties of exponents
Simplification of algebraic expressions

Suitable Grade Level

Grades 10-12