Question

If we have an expression as \[{{5}^{n+2}}=625\], then find the value of \[\sqrt[3]{n+727}\].

Answer 1

Hint: First use the property of power to expand the given equation of n. Now assume \[{{5}^{n}}\] as a variable. Now you have an equation in a variable. Find the value of the variable. Now convert the right-hand side into power of 5. By this you get a variable equal to power of 5. Substitute \[{{5}^{n}}\] back. Now you get the value of n from this equation. Substitute value of n in required term to get value of that term.

Complete step-by-step solution -
Given term in the equation is written as follows here:
\[\Rightarrow {{5}^{n+2}}=625\]
By properties of powers, we know the formula given by:
\[\Rightarrow {{a}^{b+c}}={{a}^{b}},{{a}^{c}}\]
Here a, b, c are any random integers. It is true for any number.
Here we have a = 5, b = n, c = 2 in our given term.
By substituting these values into given equation, we get it as:
\[\Rightarrow {{5}^{n}}{{.5}^{2}}=625\]
Assuming the term \[{{5}^{n}}\] as new variable t in our equation.
By substituting t into above equation, we get it as:
\[\Rightarrow t{{.5}^{2}}=625\]
So, we can write the above in other form such as:
\[\Rightarrow {{5}^{2}}t=625\]
By substituting value of \[{{5}^{2}}\], we get the equation in form:
\[\Rightarrow 25t=625\]
By dividing with 25 on both sides of equation, we see:
\[\Rightarrow t=\dfrac{625}{25}\]
By substituting value of t back into equation as:
\[\Rightarrow {{5}^{n}}=\dfrac{625}{25}\]
By writing 625 in terms of 5, we get the equation as:
\[\Rightarrow {{5}^{n}}=\dfrac{{{5}^{4}}}{25}\]
By writing 25 in terms of 5, we get the equation as:
\[\Rightarrow {{5}^{n}}=\dfrac{{{5}^{4}}}{{{5}^{2}}}\]
By basic properties of powers, we have the formula of a, b, c as:
\[\Rightarrow \dfrac{{{a}^{b}}}{{{a}^{c}}}={{a}^{b-c}}\]
By using this, we get \[{{5}^{n}}={{5}^{2}}\].
\[\Rightarrow n=2\]
By substituting this into the required term of question, we get:
\[\Rightarrow \sqrt[3]{2+727}=\sqrt[3]{729}\]
By simplifying this, we get it as: \[\sqrt[3]{2+727}=9\]
Therefore, 9 is the required value of the question.

Note: Alternatively you can do without assuming t. We are done this here just for the sake of clarity \[\sqrt[3]{729}\] is a standard value of 9. Be careful while applying properties of power because they decide the way of solution. If you make a mistake in that step the whole answer might go wrong. So, do the substituting carefully.