Simplify each expression, using exact forms.
a) [tex]$\sqrt{128}-\sqrt{32}=[/tex]
b) [tex]$\sqrt[3]{2058}+\sqrt[3]{384}=[/tex]
c) [tex]$\frac{1}{8} \sqrt[4]{24576}+\frac{1}{2} \sqrt[4]{96}=[/tex]
NOTE: To input [tex]$a \sqrt{b}$[/tex], type "a sqrt(b)". To input [tex]$a \sqrt[b]{c}$[/tex], type "a root(b)(c)".
Answer (2)
Simplify 128 and 32 to 8 2 and 4 2 , respectively, then subtract to get 4 2 .
Simplify 3 2058 and 3 384 to 7 3 6 and 4 3 6 , respectively, then add to get 11 3 6 .
Simplify 4 24576 and 4 96 to 8 4 6 and 2 4 6 , respectively, then compute 8 1 ( 8 4 6 ) + 2 1 ( 2 4 6 ) to get 2 4 6 .
Explanation
Problem Analysis We are asked to simplify three expressions involving radicals. We will simplify each radical by finding the largest perfect square, cube, or fourth power factors, and then combine like terms.
Simplifying a) a) We have 128 − 32 .
We simplify 128 as follows: 128 = 64 × 2 = 64 × 2 = 8 2 We simplify 32 as follows: 32 = 16 × 2 = 16 × 2 = 4 2 Therefore, 128 − 32 = 8 2 − 4 2 = ( 8 − 4 ) 2 = 4 2
Simplifying b) b) We have 3 2058 + 3 384 .
We simplify 3 2058 as follows: 3 2058 = 3 343 × 6 = 3 343 × 3 6 = 7 3 6 We simplify 3 384 as follows: 3 384 = 3 64 × 6 = 3 64 × 3 6 = 4 3 6 Therefore, 3 2058 + 3 384 = 7 3 6 + 4 3 6 = ( 7 + 4 ) 3 6 = 11 3 6
Simplifying c) c) We have 8 1 4 24576 + 2 1 4 96 .
We simplify 4 24576 as follows: 4 24576 = 4 4096 × 6 = 4 4096 × 4 6 = 8 4 6 We simplify 4 96 as follows: 4 96 = 4 16 × 6 = 4 16 × 4 6 = 2 4 6 Therefore, 8 1 4 24576 + 2 1 4 96 = 8 1 ( 8 4 6 ) + 2 1 ( 2 4 6 ) = 4 6 + 4 6 = 2 4 6
Final Answer a) 128 − 32 = 4 2 b) 3 2058 + 3 384 = 11 3 6 c) 8 1 4 24576 + 2 1 4 96 = 2 4 6
Examples
Radicals are used in various fields, including engineering and physics. For example, when calculating the impedance of an AC circuit, you often encounter expressions involving square roots. Simplifying these expressions allows engineers to easily determine the circuit's behavior and design appropriate components. Similarly, in physics, radicals appear in formulas for energy, momentum, and other physical quantities, making their simplification crucial for accurate calculations and predictions.
The simplified results for the expressions are: a) 4 2 , b) 11 3 6 , and c) 2 4 6 . Each radical was simplified by finding perfect factors and combining like terms. This process involves breaking down the numbers under the radicals into their prime factors and recognizing perfect squares, cubes, and fourth powers.
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