Question

How do you solve ${25^x} = \dfrac{1}{{125}}?$

Answer 1

Hint: Here we will use additive law for the power and exponent and then simplify the equations using the square of the whole square. Also, we will use the square and multiples concepts and then simplify for the required solution.

Complete step-by-step solution:
First of all take the given expression.
${25^x} = \dfrac{1}{{125}}$
By using the law of power and exponent -
${({5^2})^x} = \dfrac{1}{{{5^3}}}$
Now, using the law of inverse multiplication which states that when any term is moved from the denominator to the numerator then the sign of the power also changes. Positive power changes to negative and vice-versa.
$ \Rightarrow {({5^2})^x} = {5^{ - 3}}$
By the property of the Power rule: to raise Power to power you have to multiply the exponents such as - ${\left( {{2^a}} \right)^b} = {2^{ab}}$
$ \Rightarrow ({5^2}^x) = {5^{ - 3}}$
When bases are the same, power is taken as equal.
$ \Rightarrow 2x = ( - 3)$
When the term multiplicative on one side is moved to the opposite side then it goes to the denominator.
$ \Rightarrow x = \dfrac{{ - 3}}{2}$
This is the required solution.

Additional Information: Remember the seven basic rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules, it describes how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
i) Product of powers rule
ii) Quotient of powers rule
iii) Power of a power rule
iv) Power of a product rule
v) Power of a quotient rule
vi) Zero power rule
vii) Negative exponent rule

Note: The power is used to express mathematical equations in the short form; it is an expression that represents the repeated multiplication of the same factor. For example - $2 \times 2 \times 2$ can be expressed as ${2^3}$. Here, the number two is called the base and the exponent represents the number of times the base is used as the factor.