What is x if [8^(x+1)]^1/2=[4^(2-x)]^1/3?

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

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8^(x+1)]^1/2 = 4^(2-x)^1/3

First we will rewrite the exponents.

We know that  8= 2^3  and 4= 2^2

==> [2^3^(x+1)]^1/2 =[ 2^2^(2-x)]^1/3

Now we know that x^a^b = x^a*b

==>  [ 2^(3x+3)]^1/2 = [2^(4-2x)]^1/3

Now we will raise to the 6th power.

==> 2^(3x+3)]^3 = 2^(4-2x)^2

==> 2^(9x+9) = 2^(8-4x)

Now that the bases are equals, then, the powers are equa;.

==> 9x + 9 = 8- 4x

We will combine like terms.

==> 9x+4x = 8 -9

==> 13x = -1

==> x= -1/13

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

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What is x if [8^(x+1)]^1/2=[4^(2-x)]^1/3.

[8^(x+1)]^1/2=[4^(2-x)]^1/3.

[(2^3)^(x+1)]^(1/2) = [(2^2)^(2-x)]^(1/3), as 8 = 2^3 and 4 = 2^2.

=> (2^(3x+3)]^(1/2) = [2^(4-2x)]^(1/3), as (a^m)^n = a^(mn).

=> 2^[(3x+3)/2] = 2^[(4-2x)/3], as (a^m)^n = a^(mn).

The bases are the same. So the exponents are also equal.

=> (3x+3)/2 = (4-2x)/3.

We multiply both sides by 6.

3(3x+3) = 2(4-2x).

9x+9 = 8-4x.

9x+4x = 8-9 = -1.

13x = -1.

x = -1/13.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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First, we'll write the exponentials both sides as having matching bases.

Since 8 = 2^3 and 4 = 2^2, we'll re-write the equation:

[2^3(x+1)]^1/2=[2^2(2-x)]^1/3

2^3(x+1)/2 = 2^2(2-x)/3

Since the bases are matching, we'll apply the one to one property:

3(x+1)/2 = 2(2-x)/3

We'll cross multiply:

4(2 - x) = 9(x + 1)

8 - 4x = 9x + 9

We'll isolate x to the left side. For this reason, we'll subtract 9x  and 8 both sides:

-9x - 4x = 9 - 8

-13x = 1

x = -1/13

The solution of the equation is:

x = -1/13

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