Math  /  Algebra

Question(i+5)2×23+8=(i+5)^{2} \times 2^{3}+\sqrt{8}=

Studdy Solution

STEP 1

Assumptions
1. i i represents the imaginary unit, where i2=1 i^2 = -1 .
2. The expression to be simplified is (i+5)2×23+8(i + 5)^2 \times 2^3 + \sqrt{8}.

STEP 2

First, we need to simplify the term (i+5)2(i + 5)^2. Use the binomial expansion formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2.
(i+5)2=i2+2i5+52(i + 5)^2 = i^2 + 2 \cdot i \cdot 5 + 5^2

STEP 3

Substitute i2=1i^2 = -1 into the expression.
(i+5)2=1+2i5+25(i + 5)^2 = -1 + 2 \cdot i \cdot 5 + 25

STEP 4

Simplify the expression.
(i+5)2=1+10i+25(i + 5)^2 = -1 + 10i + 25

STEP 5

Combine like terms.
(i+5)2=24+10i(i + 5)^2 = 24 + 10i

STEP 6

Next, simplify 232^3.
23=82^3 = 8

STEP 7

Multiply the simplified (i+5)2(i + 5)^2 by 232^3.
(24+10i)×8(24 + 10i) \times 8

STEP 8

Distribute the 8 across the terms inside the parentheses.
24×8+10i×824 \times 8 + 10i \times 8

STEP 9

Calculate the products.
192+80i192 + 80i

STEP 10

Now, simplify 8\sqrt{8}. Recall that 8=4×2=4×2=22\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}.
8=22\sqrt{8} = 2\sqrt{2}

STEP 11

Combine the results from the previous steps.
192+80i+22192 + 80i + 2\sqrt{2}

STEP 12

The final simplified expression is:
192+80i+22192 + 80i + 2\sqrt{2}

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