QuestionSimplify the expression: $\sqrt{7 x^{2}} + 6 x \sqrt{175}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to simplify the expression $\sqrt{7 x^{}}+6 x \sqrt{175}$ . . We can simplify the expression by simplifying the radicals and combining like terms if possible.
3. We assume that $x$ is a real number.

STEP 2

First, we need to simplify the radical $\sqrt{7 x^{2}}$ . This can be done by taking the square root of $7x^2$ .
$\sqrt{7 x^{2}} = \sqrt{7} \cdot \sqrt{x^{2}}$

STEP 3

Now, simplify the square root of $x^2$ .
$\sqrt{x^{2}} = x$ So, $\sqrt{7 x^{2}} = x\sqrt{7}$ .

STEP 4

Next, we need to simplify the radical $\sqrt{175}$ . This can be done by factoring175 into a product of its prime factors.
$\sqrt{175} = \sqrt{^2 \cdot7}$

STEP 5

implify the square root of $5^2$ .
$\sqrt{5^2} =5$ So, $\sqrt{175} =5\sqrt{7}$ .

STEP 6

Now, substitute the simplified radicals back into the original expression.
$\sqrt{ x^{2}}+6 x \sqrt{175} = x\sqrt{} +6x(5\sqrt{})$

STEP 7

implify the expression $6x(5\sqrt{7})$ .
$6x(5\sqrt{7}) =30x\sqrt{7}$ So, $x\sqrt{7} +6x(5\sqrt{7}) = x\sqrt{7} +30x\sqrt{7}$ .

STEP 8

Combine like terms.
$x\sqrt{7} +30x\sqrt{7} = (1x +30x)\sqrt{7} =31x\sqrt{7}$ So, the simplest radical form of the expression $\sqrt{7 x^{2}}+6 x \sqrt{175}$ is $31x\sqrt{7}$ .

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