Doubling Beats Adding Even When It Starts Behind

Doubling — and it wins by an enormous margin once the steps mount up. Starting from 100, adding 50 ten times reaches 600, while doubling ten times reaches 102400.

Adding grows in a straight line; multiplying grows explosively. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: exponential form, the same-base laws, the same-index laws, and comparing the two kinds of growth.

Exponential form writes a repeated multiplication as $n^a$, where $n$ is the base and $a$ is the exponent (or index) — the number of times the base is multiplied by itself.

$$2\times 2\times 2\times 2\times 2=2^5=32$$

Here the base is 2 and the exponent is 5, read as 2 to the power 5.

More evaluations:

$$3^4=81,10^3=1000,7^2=49$$

With a negative base, brackets are essential and the sign depends on whether the exponent is even or odd:

$$(-2{)}^4=+16,(-2{)}^3=-8$$

An even exponent gives a positive result, an odd one gives a negative result.

With a fractional base, the exponent applies to numerator and denominator alike:

$${\left(\frac{2}{3}\right)}^3=\frac{2^3}{3^3}=\frac{8}{27},{\left(\frac{-3}{4}\right)}^2=\frac{9}{16}$$

Large numbers become manageable this way: a lakh is $10^5$ and a crore is $10^7$.

The brackets decide the answer, and leaving them out changes it. $(-2{)}^4=16$, because the whole of $-2$ is raised to the power, but $-2^4=-16$, because there only the 2 is raised and the minus stays outside.

Two more laws apply when the exponents match but the bases differ:

$$n^a\times m^a=(n\times m{)}^a$$
$$\frac{n^a}{m^a}={\left(\frac{n}{m}\right)}^a$$

Worked examples.

$$2^3\times 3^3=(2\times 3{)}^3=6^3=216$$

Checking directly: $8\times 27=216$. The same answer, with much smaller arithmetic.

$$\frac{6^3}{2^3}={\left(\frac{6}{2}\right)}^3=3^3=27$$

$$4^2\times 5^2=20^2=400$$

$$\frac{12^4}{4^4}=3^4=81$$

These laws are what make a fractional base work at all:

$${\left(\frac{3}{5}\right)}^2=\frac{3^2}{5^2}=\frac{9}{25}$$

So the two sets of laws cover opposite situations, and choosing between them is the skill. Same base, add or subtract the exponents. Same index, multiply or divide the bases. Trying to add exponents when the bases differ, as in writing $2^3\times 3^3=6^6$, is the standard error — the correct answer is $6^3$, because the index stays as it was.

Additive growth adds a fixed amount each step; exponential (multiplicative) growth multiplies by a fixed factor each step.

After $n$ steps, starting from $100$:

- Additive, adding 50 each time: value $=100+50n$
- Exponential, doubling each time: value $=100\times 2^n$

Working both out step by step:

- Additive: $100,150,200,250,300,350,\dots$
- Exponential: $100,200,400,800,1600,3200,\dots$

After 4 steps: additive gives $100+200=300$, exponential gives $100\times 16=1600$.

After 10 steps: additive gives

$$100+50\times 10=600$$

while exponential gives

$$100\times 2^{10}=100\times 1024=102400$$

Another example. A folded sheet of paper doubles its thickness with each fold, so after 10 folds it is $2^{10}=1024$ times as thick — which is why folding paper more than a few times becomes impossible.

And a bacterial culture that doubles every hour goes from 1 to $2^{10}=1024$ in ten hours.

The feature that surprises people is how the two cross over. In the very first steps additive can be ahead or level — after one step both reach 150 and 200, close together. Exponential growth looks slow at the start and then overtakes decisively, which is precisely why a small percentage of growth repeated many times matters far more than it first appears.