Question

How do you simplify $$2^{1000}-2^{999}$$?

Answer 1

Hint: Here, we will use the exponential identities such that we can break the first term in such a way that we will be able to take one term common from both the terms. Then we will subtract the remaining numbers and solve the equation further to get the required value.

Formula Used:
$$a^m\times a^n=a^{m+n}$$

Complete step by step solution:
We need to simplify $$2^{1000}-2^{999}$$.
We know that $$a^m\times a^n=a^{m+n}$$
Hence, we can write the first exponential term as: $$2^{1000}=2^{999}\times 2^1$$
Thus, we get,
$$2^{1000}-2^{999}=\left(2^{999}\times 2\right)-2^{999}$$
Now, taking $$2^{999}$$ common, we get,
$$\Rightarrow 2^{1000}-2^{999}=2^{999}\left(2-1\right)$$
Subtracting the terms, we get
$$\Rightarrow 2^{1000}-2^{999}=2^{999}$$

Therefore, the required value of $$2^{1000}-2^{999}$$ is $$2^{999}$$.

Additional information:
An expression that represents the repeated multiplication of the same number is known as power. Whereas, when a number is written with power then the power becomes the exponent of that particular number. It shows how many times that particular number will be multiplied by itself. Hence, whenever we are given the multiplication of the same numbers, then we can express that number with an exponent.

Note:
A very common mistake which one can do in this question is that by looking at the question $$2^{1000}-2^{999}$$, we can get it confused with the identity $${\displaystyle \frac{a^m}{a^n}}=a^m-a^n$$ which is actually not correct, the correct identity is $${\displaystyle \frac{a^m}{a^n}}=a^{m-n}$$, but due to this common mistake we can write the given expression as:
$$2^{1000}-2^{999}={\displaystyle \frac{2^{1000}}{2^{999}}}$$
And then, cancelling out the common terms from the numerator and the denominator, we get,
$$2^{1000}-2^{999}=2$$, which is incorrect
Thus, we need to be careful while solving these questions.