Math

Question Multiply the expression (7+x)2(\sqrt{7}+x)^{2} assuming all variables represent positive real numbers.

Studdy Solution

STEP 1

Assumptions
1. All variables represent positive real numbers.
2. We will use the algebraic identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 to expand the expression.

STEP 2

Identify the terms aa and bb in the algebraic identity that will be used to expand the given expression.
a=7,b=xa = \sqrt{7}, \quad b = x

STEP 3

Apply the algebraic identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 to the expression (7+x)2(\sqrt{7}+x)^{2}.

STEP 4

Calculate a2a^2, which is the square of 7\sqrt{7}.
a2=(7)2a^2 = (\sqrt{7})^2

STEP 5

Simplify a2a^2.
a2=7a^2 = 7

STEP 6

Calculate 2ab2ab, which is two times the product of 7\sqrt{7} and xx.
2ab=27x2ab = 2 \cdot \sqrt{7} \cdot x

STEP 7

Simplify 2ab2ab.
2ab=27x2ab = 2\sqrt{7}x

STEP 8

Calculate b2b^2, which is the square of xx.
b2=x2b^2 = x^2

STEP 9

Combine the results from steps 5, 7, and 8 to write the expanded form of the expression.
(7+x)2=a2+2ab+b2(\sqrt{7}+x)^{2} = a^2 + 2ab + b^2

STEP 10

Substitute a2a^2, 2ab2ab, and b2b^2 with their respective values.
(7+x)2=7+27x+x2(\sqrt{7}+x)^{2} = 7 + 2\sqrt{7}x + x^2

STEP 11

Write the final expanded expression.
(7+x)2=x2+27x+7(\sqrt{7}+x)^{2} = x^2 + 2\sqrt{7}x + 7
This is the product of the given expression.

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