Question

If ${{10}^{x}}=64$, what is the value of ${{10}^{\dfrac{x}{2}+1}}$ ?

Answer 1

Hint: Start by taking the square root of both sides of the equation given in the figure followed by multiplying the simplified equation by a factor of 10 to get the value of the required expression.

Complete step-by-step answer:
Let us start with the solution to the above question by simplifying the equation given in the question.
${{10}^{x}}=64$
Now we will take the square root of both sides of the above equation. On doing so, we get
$\sqrt{{{10}^{x}}}=\sqrt{64}$
Now we know that $\sqrt{x}$ is equal to ${{x}^{\dfrac{1}{2}}}$ . So, our equation becomes:
${{\left( {{10}^{x}} \right)}^{\dfrac{1}{2}}}=\sqrt{64}$
$\Rightarrow {{10}^{\dfrac{x}{2}}}=\sqrt{64}$
We know that the square root of 64 is equal to 8. Therefore, our equation becomes:
${{10}^{\dfrac{x}{2}}}=8$
Now we will multiply both sides of the equation with 10. On doing so, we get
${{10}^{\dfrac{x}{2}}}\times 10=8\times 10$
Now according to rule of exponent, we know that ${{10}^{\dfrac{x}{2}}}\times 10$ can be written as ${{10}^{\dfrac{x}{2}+1}}$ . So, our final equation will come out to be:
${{10}^{\dfrac{x}{2}+1}}=80$
Therefore, from the above equation, we can conclude that the value of ${{10}^{\dfrac{x}{2}+1}}$ is 80.

Note: The most crucial thing in the questions similar to the above one is the operations that you perform with the given equation so that you can convert it to a form that would give you the required results. Also, make sure that you are quite familiar with the identities related to exponents and algebra, as they are used very often.