Escribe, en tu cuaderno, el valor perteneciente a cada expresión que forman:
a) $\left[(2)^2\right]^{-1}$
b) $\left\{\left[\left(\frac{192}{15}\right)^5\right]^0\right\}^{-3}$
c) $(0,9)^{15} \div(0,9)^{12}$
d) $[4]^2$
e) $\left[\frac{\left(4^2\right)^{-1}}{16^{-9}}\right]^{-3}$
f) $\frac{\left(\frac{2}{5}\right)^{-2}}{\left(\frac{5}{2}\right)\left(\frac{5}{2}\right)} \times 2$
g) $\frac{25^{-1}}{5^{-2}}+\frac{0,05^{-2}}{0,1^{-3}}$
Answer (2)
Simplify each expression using exponent rules.
a) Apply the rule ( a m ) n = a mn to get 4 1 .
b) Use a 0 = 1 and ( a m ) n = a mn to get 1 .
c) Apply a m ÷ a n = a m − n to get 0.729 .
d) Direct calculation yields 16 .
e) Simplify using ( a m ) n = a mn and a n a m = a m − n to get 1 6 − 24 .
f) Use ( b a ) − n = ( a b ) n to get 2 .
g) Rewrite terms as fractions and use a − n = a n 1 to get 1.4 .
4 1 , 1 , 0.729 , 16 , 1 6 − 24 , 2 , 1.4
Explanation
Introduction We will simplify each expression using exponent rules and fraction arithmetic.
Simplifying a a) [ ( 2 ) 2 ] − 1 = [ 4 ] − 1 = 4 1
Simplifying b b) { [ ( 15 192 ) 5 ] 0 } − 3 = {[ 1 ] 0 } − 3 = { 1 } − 3 = 1
Simplifying c c) ( 0 , 9 ) 15 ÷ ( 0 , 9 ) 12 = ( 0.9 ) 15 − 12 = ( 0.9 ) 3 = 0.729
Simplifying d d) [ 4 ] 2 = 4 × 4 = 16
Simplifying e e) [ 1 6 − 9 ( 4 2 ) − 1 ] − 3 = [ 1 6 − 9 1 6 − 1 ] − 3 = [ 1 6 − 1 − ( − 9 ) ] − 3 = [ 1 6 8 ] − 3 = 1 6 − 24
Simplifying f f) ( 2 5 ) ( 2 5 ) ( 5 2 ) − 2 × 2 = ( 2 5 ) ( 2 5 ) ( 2 5 ) 2 × 2 = ( 2 5 ) 2 ( 2 5 ) 2 × 2 = 1 × 2 = 2
Simplifying g g) 5 − 2 2 5 − 1 + 0 , 1 − 3 0 , 0 5 − 2 = 5 − 2 ( 5 2 ) − 1 + ( 10 1 ) − 3 ( 20 1 ) − 2 = 5 − 2 5 − 2 + 1 0 3 2 0 2 = 1 + 1000 400 = 1 + 0.4 = 1.4
Final Answer The simplified expressions are: a) 4 1 b) 1 c) 0.729 d) 16 e) 1 6 − 24 f) 2 g) 1.4
Examples
Understanding and simplifying expressions with exponents is crucial in many areas of science and engineering. For example, in physics, the intensity of light decreases with the square of the distance from the source, which can be expressed using exponents. In finance, compound interest calculations involve exponents to determine the future value of an investment. These skills are also essential in computer science for analyzing algorithms and data structures.
The simplified values for the expressions are: a) 4 1 , b) 1, c) 0.729, d) 16, e) 1 6 − 24 , f) 2, g) 1.4.
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