(1) Simplify
a) $\sqrt[6]{64}$
b) $\sqrt{550}$
c) $\sqrt{\sqrt{16}}$
(2) Calculate
a) $(\sqrt{4}+3 \sqrt{3}),(\sqrt{4}-3 \sqrt{3})=$
Find the perimeter and the area of the rectangle.
Answer (2)
Simplify 6 64 to 2 .
Simplify 550 to 5 22 .
Simplify 16 to 2 .
Calculate ( 4 + 3 3 ) ( 4 − 3 3 ) to − 23 .
− 23
Explanation
Problem Analysis We are given a set of mathematical expressions to simplify and calculate. Let's break down each one step by step to ensure clarity.
Simplifying Expressions First, we need to simplify the given expressions: a) 6 64 b) 550 c) 16
Simplifying 6 64 a) To simplify 6 64 , we need to find a number that, when raised to the power of 6, equals 64. We know that 2 6 = 64 , so 6 64 = 2 .
Simplifying 550 b) To simplify 550 , we look for perfect square factors of 550. We can write 550 = 25 × 22 . Therefore, 550 = 25 × 22 = 25 × 22 = 5 22 .
Simplifying 16 c) To simplify 16 , we first find the square root of 16, which is 4. Then, we find the square root of 4, which is 2. Therefore, 16 = 4 = 2 .
Calculating the Expression Next, we need to calculate the value of the expression: a) ( 4 + 3 3 ) ( 4 − 3 3 )
Calculating ( 4 + 3 3 ) ( 4 − 3 3 ) To calculate ( 4 + 3 3 ) ( 4 − 3 3 ) , we first simplify 4 to 2. So the expression becomes ( 2 + 3 3 ) ( 2 − 3 3 ) . This is in the form of ( a + b ) ( a − b ) , which equals a 2 − b 2 . Therefore, ( 2 + 3 3 ) ( 2 − 3 3 ) = 2 2 − ( 3 3 ) 2 = 4 − ( 9 × 3 ) = 4 − 27 = − 23 .
Final Answer Therefore, the simplified expressions are: a) 6 64 = 2 b) 550 = 5 22 c) 16 = 2 And the calculated value is: a) ( 4 + 3 3 ) ( 4 − 3 3 ) = − 23
Examples
Understanding how to simplify radicals and perform algebraic manipulations is crucial in various fields, such as physics and engineering. For instance, when calculating the impedance in electrical circuits or determining the stress on structural components, simplifying radical expressions can make complex calculations more manageable. These skills are also fundamental in computer graphics for rendering images and creating realistic simulations.
The simplified expressions are: 6 64 = 2 , 550 = 5 22 , and 16 = 2 . Additionally, the product ( 4 + 3 3 ) ( 4 − 3 3 ) = − 23 .
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