Question

What is the value of n for \[\dfrac{{{2^{288}}}}{{{2^n}}} = 512\]

Answer 1

Hint: Here, we are given the expression in 2 to power something. And we need to find the value of n. We know that, \[512 = {2^9}\] and thus substitute this, in the given equation. Also, we will use dividing exponents with the same base and different exponent. So, we will use the identity \[{a^m} \div {a^n} = {a^{m - n}}\] . Thus, using this and on evaluating this, we will get the final output.

Complete step-by-step answer:
Given that,
\[\dfrac{{{2^{288}}}}{{{2^n}}} = 512\]
Exponents are used to show repeated multiplication of a number by itself.
We know that, \[512 = {2^9}\]. And so, applying this, we will get,
\[ \Rightarrow \dfrac{{{2^{288}}}}{{{2^n}}} = {2^9}\] -------- (1)
Also, let
\[{a^m} \div {a^n}\] where the exponents have different bases and the same exponent.
Here, a is the base and m and n are the exponents.
Remember that, while dividing exponential terms, if the bases are the same, we find the difference of the exponents.
\[ = \dfrac{{{a^m}}}{{{a^n}}}\]
\[ = {a^{m - n}}\]
We will use this, in above equation (1), we will get,
\[ \Rightarrow {2^{288 - n}} = {2^9}\]
If the bases of the exponents are equal in any equation then exponents must be equal, i.e.
\[{a^{p\;}} = {a^{q\;}} \Rightarrow p = q\] where a is the base and p and q are exponents.
Applying this, we will get,
\[ \Rightarrow 288 - n = 9\]
On using transposing, and moving the RHS term to LHS and LHS term ‘n’ to RHS, we will get,
\[ \Rightarrow 288 - 9 = n\]
On evaluating this, we will get,
\[ \Rightarrow 279 = n\]
\[ \Rightarrow n = 279\]
Hence, the value of n is 279.
So, the correct answer is “279”.

Note: An expression that consists of a repeated power of multiplication of the same factor is called Power/Exponent/Indices. Exponents are used to express many numbers in a single expression. Proponents follow certain rules that help in simplifying expressions which are also called its laws.
Here, a and b are non-zero, and m and n are real numbers
1) With same base:
a) \[{a^m} \times {a^n}\; = {a^{m + n}}\]
b) \[{({a^m})^n}\; = {a^{mn}}\]

2) With different base:
a) \[{a^m} \times {b^m}\; = {\left( {a \times b} \right)^m}\]
b) \[\dfrac{{{a^n}}}{{{b^n}}}\; = {\left( {\dfrac{a}{b}} \right)^n}\]