Math Problem Statement

Calculate: (Various expressions involving powers, fractions, and arithmetic operations)

Solution

Let's go through the exercises from the image you've uploaded. These are calculations involving powers, fractions, and basic operations. I'll provide step-by-step solutions for each expression.


65. Apskaičiuokite (Calculate):

a) 10(2524+21)10 - (2^5 \cdot 2^{-4} + 2^1)

  1. Calculate the powers:

    • 25=322^5 = 32
    • 24=1162^{-4} = \frac{1}{16}
    • 21=22^1 = 2
  2. Multiply and add inside the parentheses: 32116=2,2+2=432 \cdot \frac{1}{16} = 2, \quad 2 + 2 = 4

  3. Subtract: 104=610 - 4 = 6

Answer: 66


b) (32)2(52+122)(3^2)^2 - \left(5^2 + \frac{1}{2^2}\right)

  1. Calculate the powers:

    • (32)2=92=81(3^2)^2 = 9^2 = 81
    • 52=255^2 = 25
    • 122=14\frac{1}{2^2} = \frac{1}{4}
  2. Add inside the parentheses: [ 25 + \frac{1}{4} = 25.

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Math Problem Analysis

Mathematical Concepts

Exponents and Powers
Fractions
Order of Operations
Arithmetic Operations

Formulas

a^m ⋅ a^n = a^(m+n)
(a^m) / (a^n) = a^(m-n)
(a^m)^n = a^(m⋅n)

Theorems

Laws of Exponents
Basic Arithmetic Operations

Suitable Grade Level

Grades 9-11