$10$ is a base number of an exponential term and its exponent is the sum of number $2$ and half of the common logarithm of $16$. The entire exponential term is placed under a square root. The value of this radical has to find in this problem.

$\sqrt{\large 10^{\displaystyle \normalsize 2+\frac{1}{2} \log_{10} 16}}$

The exponent contains a logarithm term and the logarithmic term is multiplied by rational number $\dfrac{1}{2}$. It can be cancelled if the number of the logarithmic term is expressed in exponential notation.

Therefore, write the number $16$ in exponential notation on the basis of $4$.

$= \,\,$ $\sqrt{\large 10^{\displaystyle \normalsize 2+\frac{1}{2} \log_{10} 4^2}}$

Use power rule of logarithms to shift the power of the exponential term of the logarithm.

$= \,\,$ $\sqrt{\large 10^{\displaystyle \normalsize 2+\frac{2}{2} \log_{10} 4}}$

$= \,\,$ $\sqrt{\large 10^{\displaystyle \normalsize 2+\require{cancel} \frac{\cancel{2}}{\cancel{2}} \log_{10} 4}}$

$= \,\,$ $\sqrt{\large 10^{\displaystyle \normalsize 2+ \log_{10} 4}}$

The radicand can be split as two multiplying factors in exponential notation by the product rule of exponents.

$= \,\,$ $\sqrt{\large 10^{\displaystyle \normalsize 2} \times \large 10^{\displaystyle \normalsize \log_{10} 4}}$

The value of second exponential term is $4$ as per the fundamental logarithm identity.

$= \,\,$ $\sqrt{\large 10^{\displaystyle \normalsize 2} \times 4}$

Now, simplify the radical to obtain the answer for this problem.

$= \,\,$ $\sqrt{\large 10^{\displaystyle \normalsize 2} \times 2^2}$

$= \,\,$ $\sqrt{\large 10^{\displaystyle \normalsize 2}} \times \sqrt{\large 2^2}$

$= \,\,$ $10 \times 2$

$= \,\,$ $20$

Latest Math Topics

Latest Math Problems

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved