Question

If $ {25^{x + 1}} = \dfrac{{125}}{{{5^2}}}, $ the find the value of x

Answer 1

Hint: First of all we will convert the given expression with the common bases for the exponent and then will simplify comparing the power of the common bases and will find the value for “x”

Complete step-by-step answer:
Take the given expression: $ {25^{x + 1}} = \dfrac{{125}}{{{5^2}}} $
Find the prime factors for the composite number in the above expression. Use exponent to express the common multiples in the short form. For example: place $ 25 = 5 \times 5 = {5^2} $ and similarly for the term $ 125 = {5^5} $
Place in the above expression.
 $ {({5^2})^{x + 1}} = \dfrac{{{5^5}}}{{{5^2}}} $
Apply the power to power rule of power and exponent in the above expression which states that both the powers are multiplied.
 $ {5^{2x + 2}} = \dfrac{{{5^5}}}{{{5^2}}} $
Apply the negative exponent rule which states that when the exponent with the same base in division is written as the equivalent function taking its power in the numerator changing the sign of the powers.
 $ {5^{2x + 2}} = {5^{5 - 2}} $
Simplify the above expression finding the difference of the powers on the left hand side of the equation.
 $ {5^{2x + 2}} = {5^3} $
When bases are the same, powers are equal.
 $ 2x + 2 = 3 $
Move constant on the opposite side. When any term is moved to the opposite side then the sign of the term also changes.
 $ 2x = 3 - 2 $
Simplify –
 $ 2x = 1 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ x = \dfrac{1}{2} $
This is the required solution.
So, the correct answer is “ $ x = \dfrac{1}{2} $ ”.

Note: Always cross check the power and exponent equivalent term for any composite number since the solution completely depends on the power and exponent. Common bases on both sides of the equation have equal powers.