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Probleme propuse


  1. Determinați numerele naturale $a, b, c, d, e, f$ și $g$ din tabelul de mai jos:




Puterea		7ºc3'$f^{18}$

Baza puterii		$a$11d31

Exponentul puterii		b437g





  1. Analizați tabelul de mai jos și determinați numerele naturale $a, b, c, d, e$:




5		7$a$25$c$326099

25		49100b576d3600





  1. Scrieți sub formă de puteri cu exponent număr natural:
A$5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$;
B$12 \cdot 12 \cdot 12$;
C$7 \cdot 7 \cdot 7 \cdot 7 \cdot 7$;
D$8 \cdot 3 \cdot 8 \cdot 3 \cdot 8 \cdot 3 \cdot 8 \cdot 3$;
E$1 \cdot 1 \cdot 1 \cdot 1 \cdot 1$;

f) $\underbrace{3 \cdot 3 \cdot 3 \cdot \ldots \cdot 3}_{2017 \text{ factori}}$.


  1. Efectuați:
A$0^5 + 5^0$;
B$5^3 + 7^0$;
C$1^{200} + 0^{200} + 6^3$;
D$2^7 - 3^4$;
E$4^3 - 1^{20}$;

f) $7^3 - 4^4$;

g) $25^2 \cdot 0^0$;

h) $2^4 : 4^2$;

i) $2^3 \cdot 5^3$;

j) $5^2 \cdot 2^4$;

k) $7^2 : 5^0$;

l) $18^2 : 3^4$.



5.

AScrieți numărul 31 ca o sumă de puteri ale lui 2.
BArătați că numărul 49 se poate scrie ca o sumă de 12 pătrate ale unor numere naturale.


  1. Determinați ultima cifră a numerelor:
A$2^{2007}$;
B$3^{2017}$;
C$5^{2018}$;
D$6^{2019}$;
E$7^{2020}$;

f) $8^{2021}$;

g) $9^{2022}$;

h) $4^{2002}$.

Verified Answer :

  • Powers are calculated using the base-exponent relationship (\(a^b=P\)).
  • Arithmetic operations follow standard rules for powers, including \(x^0=1\) and \(x^1=x\).
  • Binary representations and sums of squares provide insights into number structure.
  • Cyclic patterns of last digits help determine the final digit of large powers.
  • Final results for tasks: \(a=7, b=0, c=14641; a=10, b=625, c=24, d=1024;\) various results for powers and arithmetic.

Explanation

  1. Table Analysis

Table 1 Analysis

We identify the components of the power definition \(Base^{Exponent} = Power\).

  • Column 1: \(a^b = 7^0 = 1\). Thus, \(a=7, b=0\).
  • Column 2: \(11^4 = c\). Calculating this: \(11^2 = 121\), \(121^2 = 14641\). Thus, \(c=14641\).
  • Column 3: \(d^{37} = 3^7\). This implies a non-natural number solution if taken literally as written, possibly indicating a typo in the problem source. Assuming common patterns, we define \(d=3, \text{exponent}=7\).
  • Column 4: \(31^g = f^{18}\). Without further constraints, this equation remains symbolic.

Table 2 Analysis

We examine the relationship between the top and bottom rows.

  • \(5 \to 25\) is \(5^2\).
  • \(7 \to 49\) is \(7^2\).
  • \(a \to 100\): Since \(10^2=100\), \(a=10\).
  • \(25 \to b\): Since \(25^2=625\), \(b=625\).
  • \(c \to 576\): Since \(24^2=576\), \(c=24\).
  • \(32 \to d\): Since \(32^2=1024\), \(d=1024\).
  • The pairs \((60, 3)\) and \((99, 600)\) do not follow the \(x^2\) rule, indicating they may be independent or follow a different logic not immediately apparent. We provide solutions for the consistent parts.
  1. Representing Products as Powers

Powers with Natural Exponents

Using the definition \(x \cdot x \dots \cdot x\) (\(n\) factors) = \(x^n\):

A\(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 5^6\)
B\(12 \cdot 12 \cdot 12 = 12^3\)
C\(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 = 7^5\)
D\((8 \cdot 3) \cdot (8 \cdot 3) \cdot (8 \cdot 3) \cdot (8 \cdot 3) = 24^4\)
E\(1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 = 1^5 = 1\)

f) \(\underbrace{3 \cdot 3 \dots \cdot 3}_{2017} = 3^{2017}\)

  1. Arithmetic Operations with Exponents

Arithmetic Operations

A\(0^5 + 5^0 = 0 + 1 = 1\)
B\(5^3 + 7^0 = 125 + 1 = 126\)
C\(1^{200} + 0^{200} + 6^3 = 1 + 0 + 216 = 217\)
D\(2^7 - 3^4 = 128 - 81 = 47\)
E\(4^3 - 1^{20} = 64 - 1 = 63\)

f) \(7^3 - 4^4 = 343 - 256 = 87\)
g) \(25^2 \cdot 0^0 = 625 \cdot 1 = 625\) (assuming \(0^0=1\))
h) \(2^4 : 4^2 = 16 : 16 = 1\)
i) \(2^3 \cdot 5^3 = (2 \cdot 5)^3 = 10^3 = 1000\)
j) \(5^2 \cdot 2^4 = 25 \cdot 16 = 400\)
k) \(7^2 : 5^0 = 49 : 1 = 49\)
l) \(18^2 : 3^4 = 324 : 81 = 4\)

  1. Sums and Last Digits

Sum of Powers and Squares

ABinary representation of \(31\): \(31 = 16 + 8 + 4 + 2 + 1 = 2^4 + 2^3 + 2^2 + 2^1 + 2^0\).
B\(49\) as sum of \(12\) perfect squares: We can use \(49 = 6^2 + 2^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 0^2\). Counting the squares: \(36 + 4 + 1+1+1+1+1+1+1+1+1 + 0 = 49\). That is \(12\) terms: \(1\) (of \(6^2\)), \(1\) (of \(2^2\)), \(9\) (of \(1^2\)), and \(1\) (of \(0^2\)). Total = \(1+1+9+1 = 12\).

Last Digit Identification

A\(2^{2007}\): Cycle of \(2^n \pmod{10}\) is \((2, 4, 8, 6)\). \(2007 \div 4 = 501\) remainder \(3\). The 3rd in the cycle is \(8\).
B\(3^{2017}\): Cycle is \((3, 9, 7, 1)\). \(2017 \div 4 = 504\) remainder \(1\). The 1st is \(3\).
C\(5^{2018}\): Always ends in \(5\).
D\(6^{2019}\): Always ends in \(6\).
E\(7^{2020}\): Cycle \((7, 9, 3, 1)\). \(2020 \div 4 = 505\) remainder \(0\). The 4th is \(1\).

f) \(8^{2021}\): Cycle \((8, 4, 2, 6)\). \(2021 \div 4 = 505\) remainder \(1\). The 1st is \(8\).
g) \(9^{2022}\): Cycle \((9, 1)\). \(2022 \div 2 = 1011\) remainder \(0\). The 2nd is \(1\).
h) \(4^{2002}\): Cycle \((4, 6)\). \(2002 \div 2 = 1001\) remainder \(0\). The 2nd is \(6\).

Examples

Understanding exponents is vital in real-world scenarios such as calculating compound interest, where money grows exponentially, or in computer science, where binary representations like \(31 = 2^4 + 2^3 + 2^2 + 2^1 + 2^0\) are used to process data. These concepts help in optimizing storage, managing financial growth, and solving complex algorithmic problems efficiently.

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