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# Solve. (6√6)^−x+3 = 1/6·216^2x−3

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Solve. (6√6)^−x+3 = 1/6·216^2x−3

Jan 14, 2020

#1
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Is it [(6√6)^-x]+3=1/6·[216^2x]−3

or (6√6)^[−x+3] = 1/6·216^[2x−3]

Jan 14, 2020
#2
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it's  (6√6)^[−x+3] = 1/6·216^[2x−3]

Guest Jan 14, 2020
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This problem may look it involves logs but It can be solved using exponentials and indices laws.

Let's first write it in latex form so it is clearer:

$$(6*\sqrt{6})^{-x+3}=\frac{1}{6}*216^{2x-3}$$

Notice the bracket $$(6*\sqrt{6})^{-x+3}$$ Ignore the exponent right now,

we know that: $$(6*\sqrt{6})$$=$$(6*6^{0.5})$$

Now back the exponent

$$(6*6^{0.5})^{-x+3}$$ using the law: $$(a*b)^c=a^c*b^c$$

$$6^{-x+3}*6^{0.5*(-x+3)}$$ Multiply the bracket by the half, we get, $$6^{-x+3}*6^{-0.5x+1.5}$$

Using the law $$a^{b+c}=a^b*a^c$$

$$6^{-x}*6^3*6^{-0.5x}*6^{1.5}$$

The right hand side now,

$$\frac{1}{6}*216^{2x-3}$$

We know that

$$216=6^3$$

$$\frac{1}{6}=6^{-1}$$

$$6^{-1}*6^{3(2x-3)}$$ apply the same rules,

$$6^{-1}*6^{6x-9}$$

Now we have both sides simplified

$$6^{-x}*6^3*6^{-0.5x}*6^{1.5}$$$$=$$$$6^{-1}*6^{6x-9}$$

Notice -x and -0.5x same base, we add exponents

Notice 3 and 1.5, same base again so we add exponents

$$6^{-1.5x}*6^{4.5}$$$$=$$$$6^{-1}*6^{6x-9}$$

Now divide by $$6^{-1}$$

$$6^{-1.5x}*6^{4.5-(-1)}$$$$=$$$$6^{-1.5x}*6^{5.5}=6^{6x-9}$$

Notice $$6^{6x-9}=6^{6x}*6^{-9}$$

$$6^{-1.5x}*6^{5.5}=6^{6x}*6^{-9}$$

Divide by $$6^{-9}$$

$$6^{-1.5x}*6^{14.5}=6^{6x}$$

Divide by $$6^{-1.5x}$$

$$6^{14.5}=6^{7.5x}$$

The same base, therefore, exponents are equal (Law: $$a^b=a^c,b=c$$)

$$14.5=7.5x$$ Divide by 7.5

$$x=\frac{14.5}{7.5}=1.93333333$$ the 3 is recurring.

Jan 14, 2020