e) [tex]\left[\frac{\left(4^2\right)^{-1}}{16^{-9}}\right]^{-3}[/tex]
f) [tex]\frac{\left(\frac{2}{5}\right)^{-2}}{\left(\frac{5}{2}\right)\left(\frac{5}{2}\right)} \times 2[/tex]
g) [tex]\frac{25^{-1}}{5^{-2}}+\frac{0,05^{-2}}{0,1^{-3}}[/tex]
Answer (2)
Simplify expression e) using exponent rules: [ 1 6 − 9 ( 4 2 ) − 1 ] − 3 = 2 96 1 .
Simplify expression f) using exponent rules: ( 2 5 ) ( 2 5 ) ( 5 2 ) − 2 × 2 = 2 .
Simplify expression 9) by rewriting decimals as fractions and using exponent rules: 5 − 2 2 5 − 1 + 0 , 1 − 3 0 , 0 5 − 2 = 5 7 .
The simplified expressions are 2 96 1 , 2 , 5 7 .
Explanation
Problem Setup We are given three expressions to simplify: e) [ 1 6 − 9 ( 4 2 ) − 1 ] − 3 f) ( 2 5 ) ( 2 5 ) ( 5 2 ) − 2 × 2
5 − 2 2 5 − 1 + 0 , 1 − 3 0 , 0 5 − 2
Simplifying Expression e Let's simplify expression e) first. We will use the exponent rules ( a m ) n = a mn and a n a m = a m − n .
[ 1 6 − 9 ( 4 2 ) − 1 ] − 3 = [ 1 6 − 9 4 − 2 ] − 3 = [ ( 2 4 ) − 9 ( 2 2 ) − 2 ] − 3 = [ 2 − 36 2 − 4 ] − 3 = [ 2 − 4 − ( − 36 ) ] − 3 = [ 2 32 ] − 3 = 2 − 96 = 2 96 1
Simplifying Expression f Now, let's simplify expression f). We will use the exponent rule ( b a ) − n = ( a b ) n .
( 2 5 ) ( 2 5 ) ( 5 2 ) − 2 × 2 = ( 2 5 ) 2 ( 2 5 ) 2 × 2 = 1 × 2 = 2
Simplifying Expression 9 Finally, let's simplify expression 9). We will rewrite decimals as fractions and use exponent rules.
5 − 2 2 5 − 1 + 0 , 1 − 3 0 , 0 5 − 2 = 5 − 2 ( 5 2 ) − 1 + ( 10 1 ) − 3 ( 100 5 ) − 2 = 5 − 2 5 − 2 + ( 10 1 ) − 3 ( 20 1 ) − 2 = 1 + ( 10 ) 3 ( 20 ) 2 = 1 + 1000 400 = 1 + 5 2 = 5 7
Final Answer Therefore, the simplified expressions are: e) 2 96 1 f) 2
5 7
Examples
Understanding and simplifying exponential expressions is crucial in various fields, such as computer science (analyzing algorithm complexity), physics (describing radioactive decay), and finance (calculating compound interest). For instance, in computer science, the time complexity of an algorithm might be expressed as O ( 2 n ) , where n is the input size. Simplifying such expressions helps in comparing the efficiency of different algorithms. Similarly, in finance, understanding exponential growth is essential for making informed investment decisions.
The simplified expressions are: e) 2 96 1 , f) 2, and g) 5 7 . Each expression was simplified using exponent rules and properties of fractions. This process highlighted how to manipulate exponents and simplify complex expressions step by step.
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