Question

2^x*4^x=8^1/3*32^1/5 find value of x

Answer 1

Hey there!

Given equation to find the value of x,  

$\bf{2^x \times 4^x = 8^{\frac{1}{3}} \times 32^{\frac{1}{5}}}$.

Take 1/3 as a cube root over the value of "8".

$\bf{2^x \times 4^x = \sqrt[3]{8} \times 32^{\frac{1}{5}}}$.

Now, apply the rules for exponential variables that is,

$\bf{a^c b^c = (ab)^c}$

Here,  

$\bf{2^x \times 4^x = (2 \times 4)^x = 8^x}$

$\bf{8^x = \sqrt[3]{8} \times 32^{\frac{1}{5}}}$

$\bf{8^x = 2 \times \sqrt[5]{32}}$

$\bf{8^x = 2 \times 2}$

$\bf{8^x = 4}$

Now, convert the following value of "4" into $\bf{2^2}$ or to the base of 2 and on power of 2.

$\bf{8^x = 2^2}$

Converting the base of $\bf{8^x}$ with respect to "2" and powering by "3" that is,

$\bf{8^x = (2^3)^x}$

$\bf{(2^3)^x = 2^2}$

Now, by applying the rule of exponents to the values of it that is,

$\bf{(a^b)^c = a^{b \times c}}$

Here,

$\bf{(2^3)^x = 2^{3 \times x}}$

$\bf{2^{3 \times x} = 2^2}$

Apply the function of "x" rule to get the final value for this query that is,

$\bf{a^{f(x)} = a^{g(x)}}$

Then we can eliminate the same bases to get as,

$\bf{f(x) = g(x)}$

Therefore,

$\bf{3x = 2}$

Divide both the sides with a denominator value of "3".

$\bf{\frac{3x}{3} = \frac{2}{3}}$

Simplify the given values by cancelling similar values on L.H.S.

$\boxed{\bf{\therefore \: \: x = \frac{2}{3}}}$

Which is the final value or the required answer for this type of query.

Hope this helps you and solves your doubts of solving exponents for finding the values for "x"!!!!