Drag each expression to the correct location on the table. Simplify each exponential expression using the properties of exponents and match it to the correct answer.
[tex]3^2 \cdot\left(3^3\right)^2 \cdot 3^{-8}[/tex] [tex]\frac{\left(3^2\right)(2 \cdot 3)^{-3}}{2^{-3}}[/tex] [tex]\frac{\left(2^{-1}\right)(3 \cdot 2)^4}{(3 \cdot 2)^3}[/tex] [tex]2^5 \cdot 3^5 \cdot 6^{-5}[/tex] [tex]\frac{\left(2^3\right)(2 \cdot 3)^{-1}}{2^2}[/tex]
| 3 | 1 | [tex]\frac{1}{3}[/tex] |
|---|---|---|
| | | |
Answer (2)
Simplify 3 2 c d o t ( 3 3 ) 2 c d o t 3 − 8 to 3 2 + 6 − 8 = 3 0 = 1 .
Simplify 2 − 3 ( 3 2 ) ( 2 c d o t 3 ) − 3 to 3 2 − 3 c d o t 2 − 3 + 3 = 3 − 1 = 3 1 .
Simplify ( 3 c d o t 2 ) 3 ( 2 − 1 ) ( 3 c d o t 2 ) 4 to 2 − 1 + 4 − 3 c d o t 3 4 − 3 = 3 .
Simplify 2 5 c d o t 3 5 c d o t 6 − 5 to 2 5 − 5 c d o t 3 5 − 5 = 1 .
Simplify 2 2 ( 2 3 ) ( 2 c d o t 3 ) − 1 to 2 3 − 1 − 2 c d o t 3 − 1 = 3 1 .
The simplified expressions are 1 , 3 1 , 3 , 1 , 3 1 .
Explanation
Problem Analysis We are given five exponential expressions to simplify and match to one of the values: 3, 1, or 1/3. Let's simplify each expression step by step using the properties of exponents.
Simplifying the First Expression The first expression is 3 2 c d o t ( 3 3 ) 2 c d o t 3 − 8 . Using the power of a power rule, ( a m ) n = a m c d o t n , we have ( 3 3 ) 2 = 3 3 c d o t 2 = 3 6 . Thus, the expression becomes 3 2 c d o t 3 6 c d o t 3 − 8 . Using the product of powers rule, a m c d o t a n = a m + n , we have 3 2 + 6 − 8 = 3 0 . Since any non-zero number raised to the power of 0 is 1, we get 3 0 = 1 .
Simplifying the Second Expression The second expression is 2 − 3 ( 3 2 ) ( 2 c d o t 3 ) − 3 . First, we simplify ( 2 c d o t 3 ) − 3 as 2 − 3 c d o t 3 − 3 . Thus, the expression becomes 2 − 3 3 2 c d o t 2 − 3 c d o t 3 − 3 . We can rewrite this as 3 2 − 3 c d o t 2 − 3 − ( − 3 ) = 3 − 1 c d o t 2 0 = 3 − 1 c d o t 1 = 3 1 .
Simplifying the Third Expression The third expression is ( 3 c d o t 2 ) 3 ( 2 − 1 ) ( 3 c d o t 2 ) 4 . We can rewrite this as 3 3 c d o t 2 3 2 − 1 c d o t 3 4 c d o t 2 4 . This simplifies to 2 − 1 + 4 − 3 c d o t 3 4 − 3 = 2 0 c d o t 3 1 = 1 c d o t 3 = 3 .
Simplifying the Fourth Expression The fourth expression is 2 5 c d o t 3 5 c d o t 6 − 5 . We can rewrite 6 − 5 as ( 2 c d o t 3 ) − 5 = 2 − 5 c d o t 3 − 5 . Thus, the expression becomes 2 5 c d o t 3 5 c d o t 2 − 5 c d o t 3 − 5 . This simplifies to 2 5 − 5 c d o t 3 5 − 5 = 2 0 c d o t 3 0 = 1 c d o t 1 = 1 .
Simplifying the Fifth Expression The fifth expression is 2 2 ( 2 3 ) ( 2 c d o t 3 ) − 1 . We can rewrite ( 2 c d o t 3 ) − 1 as 2 − 1 c d o t 3 − 1 . Thus, the expression becomes 2 2 2 3 c d o t 2 − 1 c d o t 3 − 1 . This simplifies to 2 3 − 1 − 2 c d o t 3 − 1 = 2 0 c d o t 3 − 1 = 1 c d o t 3 1 = 3 1 .
Matching the Expressions Now, let's match the simplified expressions to the correct values:
3 2 c d o t ( 3 3 ) 2 c d o t 3 − 8 = 1
2 − 3 ( 3 2 ) ( 2 c d o t 3 ) − 3 = 3 1
( 3 c d o t 2 ) 3 ( 2 − 1 ) ( 3 c d o t 2 ) 4 = 3
2 5 c d o t 3 5 c d o t 6 − 5 = 1
2 2 ( 2 3 ) ( 2 c d o t 3 ) − 1 = 3 1
Examples
Exponential expressions are used in various fields such as finance, physics, and computer science. For example, in finance, compound interest is calculated using exponential growth. In physics, radioactive decay is modeled using exponential decay. In computer science, the efficiency of algorithms is often expressed using exponential notation. Understanding how to simplify exponential expressions is crucial for solving problems in these fields.
The expressions simplify to 1, \frac{1}{3}, and 3 in a clearly defined way. The simplified forms match up as: 1 with two expressions, \frac{1}{3} with two others, and 3 with one. Thus, each matches correctly to form an organized table of results.
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