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How do you solve for \[x\] in\[{{\left( 125 \right)}^{x}}=625\]?

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint:In the given question, we have been asked to find the value of ‘x’ and it is given that\[{{\left( 125 \right)}^{x}}=625\]. In order to solve the question, first we write the numbers in exponential form. Then we need to use the property of product of power which states that if there is a power to the power, then we multiply the power such that\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]. After that we will equate the powers with each other and solve in a way we solve the general equation.

Formula used:
The property of product of power which states that if there is a power to the power, then we multiply the power such that\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\].

Complete step by step solution:
We have given that,
\[\Rightarrow {{\left( 125 \right)}^{x}}=625\]
As we know that,
\[\Rightarrow 5\times 5\times 5={{5}^{3}}=125\] And
\[\Rightarrow 5\times 5\times 5\times 5={{5}^{4}}=625\]
Substituting \[125={{5}^{3}}\] and \[625={{5}^{4}}\] in the given equation, we get
\[\Rightarrow {{\left( {{5}^{3}} \right)}^{x}}={{5}^{4}}\]
Using the property of product of a power, i.e.
If there is a power to the power, then we multiply the power.
\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]
Applying the property in the equation, we get
\[\Rightarrow {{5}^{3x}}={{5}^{4}}\]
Now both the sides base are equal i.e. 5.
Equation powers with each other, we get
\[\Rightarrow 3x=4\]
Now solving for the value of ‘x’.
Dividing both the sides of the equation by 3, we get
\[\Rightarrow \dfrac{3x}{3}=\dfrac{4}{3}\]
Simplifying the above, we get
\[\Rightarrow x=\dfrac{4}{3}\]
Therefore, the value of \[x=\dfrac{4}{3}\] is the required answer.

Note: While solving these types of questions, students need to remember the properties of exponent and powers. For converting the number into exponential form, we will need to find the prime factorization of that number and we will get our number in exponential form. Exponents are used to write the same number in multiplication form using a simple format.
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