Math Problem Statement
Diketahui a = 3 dan b = 2. Bilangan berpangkat (4/9)^3 × (3/16)^-2 jika dinyatakan dalam a dan b adalah…
Solution
This problem requires simplifying the given expression and rewriting it in terms of and .
Given:
where and . Let's express the powers of and in terms of and :
Step 1: Rewrite the terms
\quad \text{and} \quad \frac{3}{16} = \frac{a}{b^4}.$$ #### Step 2: Apply powers 1. $$\left( \frac{4}{9} \right)^3 = \left( \frac{b^2}{a^2} \right)^3 = \frac{b^6}{a^6}$$. 2. $$\left( \frac{3}{16} \right)^{-2} = \left( \frac{a}{b^4} \right)^{-2} = \frac{b^8}{a^2}$$. #### Step 3: Multiply the results $$\frac{b^6}{a^6} \times \frac{b^8}{a^2} = \frac{b^{6+8}}{a^{6+2}} = \frac{b^{14}}{a^8}.$$ #### Step 4: Final result Rewriting the answer in the given options: $$\frac{b^{14}}{a^8}.$$ This matches **option D** in the problem. --- Let me know if you'd like further clarification or a breakdown of any step! Here are some related questions: 1. How do we simplify fractions with powers involving variables? 2. Why does raising a fraction to a negative power involve reciprocal operations? 3. How do we express numbers like $$4, 9, 16$$ in terms of given variables? 4. How do the laws of exponents apply when multiplying powers? 5. Can you explain another way to check the solution? **Tip:** Always break down numbers into their prime factorization to simplify problems like this.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponentiation
Fraction simplification
Algebraic expressions
Formulas
a^m × a^n = a^(m+n)
(a/b)^m = a^m / b^m
(a/b)^-n = (b/a)^n
Theorems
Properties of exponents
Simplification of fractional powers
Suitable Grade Level
Grades 10-11