Question

How do you solve ${10^{x - 3}} = {100^{4x - 5}}$

Answer 1

Hint: This problem comes under powers and exponents. Power is the multiplication of the same factor. Exponent is the number of times the base is used as a factor. Here we compare both sides of the given exponents and then with a basic method of solving squares we simplify the given powers and exponents. And then with a basic algebraic method of solving we will find the value of $x$. Then with basic mathematical calculation and complete step by step explanation.

Complete step-by-step solution:
Consider the equation
$ \Rightarrow {10^{x - 3}} = {100^{4x - 5}}$
Now convert 100 in powers in 10 in right hand side, we know that ${10^2} = 100$ by applying this in above equation we get
$ \Rightarrow {10^{x - 3}} = {10^{2(4x - 5)}}$
Now multiply exponent’s powers, we get
$ \Rightarrow {10^{x - 3}} = {10^{8x - 10}}$
Since the base of both sides of expression is same we compare the exponents value, we get
$ \Rightarrow x - 3 = 8x - 10$
Now separating variable and numerals, then we get
$ \Rightarrow 10 - 3 = 8x - x$
Let us subtract the term and we get
$ \Rightarrow 7 = 7x$
Now divide seven on both sides, we obtain the value of $x$
$ \Rightarrow x = 1$
Thus we solve the expression.

Hence the required answer $x = 1$

Note: The expressions that represent repeated multiplication of the same factor is power. The exponents correspond to the number of times the base is used as factors. The power values have to be simplified by square of the number after converting square then we obtain the same base value then comparing exponents we solve by algebraic method to find the solution. This also makes a simple concept making the method simple on basis of powers and exponents. Then needs to be strong on basic mathematical calculation and substitutions.