Question

State whether the given statement is true or false and justify your answer:
${{3}^{0}}={{1000}^{0}}$

Answer 1

Hint: First, find how a number will behave if we raise it to the power of zero. Suppose you have a number ${{x}^{y}}$. Divide it by itself to give $\dfrac{{{x}^{y}}}{{{x}^{y}}}={{x}^{y-y}}={{x}^{0}}$. This is equal to 1 because when a number is divided by itself, the result is 1. Using the fact above to find the values of LHS and the RHS and compare them to find if the given statement is true or false.

Complete step-by-step answer:
In this question, we need to state whether the given statement: ${{3}^{0}}={{1000}^{0}}$ is true or false and justify the answer we arrive at.
We first need to find out how a number will behave if we raise it to the power of zero.
We know that any number raised to the power zero gives 1 as the result. This is known as the zero power rule.
We arrive at this fact in the following way:
Let as suppose, we have a number x raised to the power y, i.e. ${{x}^{y}}$ where x and y are positive integers.
Now, let us divide this number by itself. Doing this we will get the following: $\dfrac{{{x}^{y}}}{{{x}^{y}}}$
Now, we know that when a number x to the power of a is divided by x to the power of b, we get x to the power of the difference of a and b.
i.e. $\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$
Using this property above on $\dfrac{{{x}^{y}}}{{{x}^{y}}}$, we will get the following:
$\dfrac{{{x}^{y}}}{{{x}^{y}}}={{x}^{y-y}}={{x}^{0}}$
Also, we know that when a number is divided by itself, the result is 1.
So, $\dfrac{{{x}^{y}}}{{{x}^{y}}}=1$
Hence, \[\dfrac{{{x}^{y}}}{{{x}^{y}}}=1={{x}^{0}}\]
That’s why any number raised to the power of zero gives 1 as the result.
Using this in our question, we have the following:
LHS = \[{{3}^{0}}=1\]
RHS = \[{{1000}^{0}}=1\]
Hence, LHS = RHS
Therefore, ${{3}^{0}}={{1000}^{0}}$ is a true statement.

Note: In this question, it is very important to know and understand the fact that any number raised to the power of zero gives 1 as the result, i.e. ${{x}^{0}}=1$. This is called the zero power rule.