Math Problem Statement

Pangkat positif dari \(\left( \frac{a^3 b^{-1}}{c^{-1} d^4} \right)^{-2}\) adalah ....

Solution

Let's analyze the mathematical expression provided in the image and simplify it.

The given expression is:

(a3b1c1d4)2\left( \frac{a^3 b^{-1}}{c^{-1} d^4} \right)^{-2}

Step-by-Step Simplification:

  1. Simplify the exponents inside the parentheses: The expression inside the parentheses is a3b1c1d4\frac{a^3 b^{-1}}{c^{-1} d^4}.

    • The negative exponent of bb means b1=1bb^{-1} = \frac{1}{b}.
    • The negative exponent of cc means c1=1cc^{-1} = \frac{1}{c}.

    This transforms the expression to:

    a31b1cd4=a3bcd4=a3cbd4\frac{a^3 \cdot \frac{1}{b}}{\frac{1}{c} \cdot d^4} = \frac{a^3}{b} \cdot \frac{c}{d^4} = \frac{a^3 c}{b d^4}

  2. Apply the exponent -2 to the whole expression: Now, we need to apply the exponent of 2-2 to the entire fraction a3cbd4\frac{a^3 c}{b d^4}.

    Using the power rule (xy)n=xnyn\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}, we apply the exponent 2-2 to both the numerator and denominator:

    (a3cbd4)2=(a3c)2(bd4)2\left( \frac{a^3 c}{b d^4} \right)^{-2} = \frac{\left( a^3 c \right)^{-2}}{\left( b d^4 \right)^{-2}}

    Simplifying both the numerator and denominator:

    • (a3c)2=a6c2(a^3 c)^{-2} = a^{-6} c^{-2}
    • (bd4)2=b2d8(b d^4)^{-2} = b^{-2} d^{-8}

    So the expression becomes:

    a6c2b2d8=a6c21b2d8=a6c2b2d8\frac{a^{-6} c^{-2}}{b^{-2} d^{-8}} = a^{-6} c^{-2} \cdot \frac{1}{b^{-2} d^{-8}} = a^{-6} c^{-2} b^2 d^8

  3. Convert to positive exponents: Now, to express everything with positive exponents, we get:

    b2d8a6c2=b2d8a6c2b^2 d^8 a^{-6} c^{-2} = \frac{b^2 d^8}{a^6 c^2}

Thus, the simplified expression with all positive exponents is:

b2d8a6c2\boxed{\frac{b^2 d^8}{a^6 c^2}}

Would you like further details or have any questions?

Here are 5 related questions to explore:

  1. How do you apply negative exponents to a fraction?
  2. What is the general rule for simplifying an expression with multiple terms raised to a negative exponent?
  3. How does changing the exponent sign affect the position of variables in the numerator or denominator?
  4. How do you simplify expressions with both positive and negative exponents?
  5. How does the power of a product rule work when applying exponents?

Tip: When simplifying expressions with exponents, always manage the signs carefully and apply exponent rules step by step to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Exponents
Negative Exponents
Simplifying Fractions with Exponents

Formulas

Power of a quotient rule: \(\left( \frac{x}{y} \right)^n = \frac{x^n}{y^n}\)
Negative exponent rule: \(x^{-n} = \frac{1}{x^n}\)

Theorems

Exponentiation rules
Power of a fraction

Suitable Grade Level

Grades 10-12