Simplify $\frac{9^{2 x+3} \div 27^{4 x-7}}{81^x \times 3^{x+1}}$
Answer (2)
Express all terms as powers of 3.
Simplify the exponents using the power of a power rule.
Use the properties of exponents to simplify the division and multiplication.
Simplify the resulting fraction to obtain the final expression: 3 − 13 x + 26
Explanation
Problem Analysis We are asked to simplify the expression 8 1 x × 3 x + 1 9 2 x + 3 ÷ 2 7 4 x − 7 . To do this, we will express each term as a power of 3 and then use the properties of exponents to simplify the expression.
Expressing as Powers of 3 First, we express each base as a power of 3: 9 = 3 2 27 = 3 3 81 = 3 4
Rewriting the Expression Now, we rewrite the expression using these powers of 3: 8 1 x × 3 x + 1 9 2 x + 3 ÷ 2 7 4 x − 7 = ( 3 4 ) x × 3 x + 1 ( 3 2 ) 2 x + 3 ÷ ( 3 3 ) 4 x − 7
Simplifying Exponents Next, we simplify the exponents: ( 3 4 ) x × 3 x + 1 ( 3 2 ) 2 x + 3 ÷ ( 3 3 ) 4 x − 7 = 3 4 x × 3 x + 1 3 2 ( 2 x + 3 ) ÷ 3 3 ( 4 x − 7 ) = 3 4 x × 3 x + 1 3 4 x + 6 ÷ 3 12 x − 21
Simplifying Division and Multiplication Now, we use the properties of exponents to simplify the division and multiplication: 3 4 x × 3 x + 1 3 4 x + 6 ÷ 3 12 x − 21 = 3 4 x + x + 1 3 ( 4 x + 6 ) − ( 12 x − 21 ) = 3 5 x + 1 3 4 x + 6 − 12 x + 21 = 3 5 x + 1 3 − 8 x + 27
Simplifying the Fraction Finally, we simplify the fraction: 3 5 x + 1 3 − 8 x + 27 = 3 ( − 8 x + 27 ) − ( 5 x + 1 ) = 3 − 8 x + 27 − 5 x − 1 = 3 − 13 x + 26 Thus, the simplified expression is 3 − 13 x + 26 .
Evaluating at x=2 If x = 2 , then the expression becomes 3 − 13 ( 2 ) + 26 = 3 − 26 + 26 = 3 0 = 1 .
Final Answer Therefore, the simplified expression is 3 − 13 x + 26 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, including physics and computer science. For example, in physics, you might use these skills to simplify equations describing radioactive decay or the behavior of electrical circuits. In computer science, you might use them to analyze the efficiency of algorithms or to understand data compression techniques. Suppose you are analyzing the growth rate of a population of bacteria, where the population size at time t is given by P ( t ) = P 0 \tmes 2 k t , where P 0 is the initial population size and k is a constant. Simplifying such exponential expressions helps in predicting population sizes at different times.
The expression 8 1 x × 3 x + 1 9 2 x + 3 ÷ 2 7 4 x − 7 simplifies to 3 − 13 x + 26 . This simplification involves rewriting all terms as powers of 3 and applying the properties of exponents. Following these steps leads to the final result.
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