What is the value of $3 x \sqrt{8 x} - 4 \sqrt{x^{3}} + 3 x \sqrt{72 x}$ $3xsqrt(8x) - 4sqrt(x^3) + 3xsqrt(72x)$ when $x = 2$ $x = 2$ ?

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What is the value of $3 x \sqrt{8 x} - 4 \sqrt{x^{3}} + 3 x \sqrt{72 x}$ $3xsqrt(8x) - 4sqrt(x^3) + 3xsqrt(72x)$ when $x = 2$ $x = 2$ ?

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Answer 1 · 2016-01-01T18:43:45

$96 - 8 \sqrt 2 \approx 84.7$ $96 - 8 √2 ≈ 84.7$ ( $1$ $1$ decimal place)

Explanation:

Substitute $x = 2$ $x =2$ into expression:
=6√16 - 4√8 + 6√144
=24 - 8√2 + 72
= 96 - 8√2

Answer 2 · 2016-01-01T20:37:00

$As an approximate decimal value this is 84.69 to 2 decimal places.$ $color(brown)("As an approximate decimal value this is "84.69" to 2 decimal places.")$

$To an EXACT value this is \Rightarrow 96 - 8 \sqrt{2}$ $color(green)("To an EXACT value this is "=>96 - 8 sqrt(2))$

Hopefully my logic has not gone wrong!

Explanation:

Not taking any shortcuts so you can see what is happening.

Given: $3 x \sqrt{8 x} - 4 \sqrt{x^{3}} + 3 x \sqrt{72 x} . .$ $3xsqrt(8x) - 4 sqrt(x^3)+3xsqrt(72x)color(white)(..)$ ...... (1)

We may choose two approaches.

$Approach 1$ $color(blue)("Approach 1")$

Just do a strait substitution and calculate. This will most likely yield an imprecise decimal solution. Note they say "what is the value". Not "what is the approximate value".

$Approach 2$ $color(blue)("Approach 2")$

Simplify and stick with surds so that any answer is guaranteed to be precise.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Adopting approach 2$ $color(blue)("Adopting approach 2")$
Looking for factors that can be used to simplify:

Notice that $9 \times 8 = 72$ $9xx8 =72$ so we have a similarity within the expression of $8 x$ $8x$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Consider: - 4 \sqrt{x^{3}}$ $color(brown)("Consider: "-4sqrt(x^3) )$ If you take the 4 inside the root you have to square it.

Write as : $- \sqrt{16 x \times x^{2}} = - \sqrt{8 x \times 2 x^{2}}$ $-sqrt(16x xx x^2)=-sqrt(8x xx 2x^2)$

$\equiv - (x \sqrt{2} \times \sqrt{8 x})$ $-=-(xsqrt(2) xx sqrt(8x))$ ........................(2)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Consider: + 3 x \sqrt{72 x}$ $color(brown)("Consider: "+3xsqrt(72x) )$

Write as: $+ 3 x \sqrt{9 \times 8 x} \equiv + 9 x \sqrt{8 x}$ $+3x sqrt(9 xx8x) -= +9xsqrt(8x)$ ......(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Substitute (2 ) and (3 ) into (1 ) giving:$ $color(blue)("Substitute (2 ) and (3 ) into (1 ) giving:")$

$3 x \sqrt{8 x} - (x \sqrt{2} \times \sqrt{8 x}) + 9 x \sqrt{8 x}$ $3xsqrt(8x) -(xsqrt(2) xx sqrt(8x))+9xsqrt(8x)$ ........(4)

$Factoring out the \sqrt{8 x}$ $color(brown)("Factoring out the "sqrt(8x))$

$\sqrt{8 x} . (3 x - \sqrt{2} . x + 9 x)$ $sqrt(8x)color(white)(.)( 3x - sqrt(2 )color(white)(.)x+9x)$

$\sqrt{8 x} . (12 x - \sqrt{2} . x)$ $sqrt(8x)color(white)(.)( 12x - sqrt(2 )color(white)(.)x)$

$Factoring out the x from the btackets$ $color(brown)("Factoring out the "x " from the btackets")$

$x \sqrt{8 x} (12 - \sqrt{2})$ $xsqrt(8x)(12-sqrt(2))$ .........................(5)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Substitute x = 2 in (5 )$ $color(blue)("Substitute "x=2 " in (5 )")$

$(2) (\sqrt{16}) (12 - \sqrt{2})$ $(2)(sqrt(16))(12-sqrt(2))$

$8 (12 - \sqrt{2})$ $8(12-sqrt(2))$

$\Rightarrow 96 - 8 \sqrt{2}$ $=>96 - 8 sqrt(2)$

$As an approximate decimal value this is 84.69 to 2 decimal places.$ $color(brown)("As an approximate decimal value this is "84.69" to 2 decimal places.")$

$To an EXACT value this is \Rightarrow 96 - 8 \sqrt{2}$ $color(green)("To an EXACT value this is "=>96 - 8 sqrt(2))$

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What is the value of $3 x \sqrt{8 x} - 4 \sqrt{x^{3}} + 3 x \sqrt{72 x}$ $3xsqrt(8x) - 4sqrt(x^3) + 3xsqrt(72x)$ when $x = 2$ $x = 2$ ?

2 Answers

Explanation:

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What is the value of 3x√8x−4√x3+3x√72x3xsqrt(8x) - 4sqrt(x^3) + 3xsqrt(72x) when x=2x = 2?

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What is the value of $3 x \sqrt{8 x} - 4 \sqrt{x^{3}} + 3 x \sqrt{72 x}$ $3xsqrt(8x) - 4sqrt(x^3) + 3xsqrt(72x)$ when $x = 2$ $x = 2$ ?