Anything Raised to the Power Zero Equals One

Because dividing a power by itself must give 1, and the division law makes the index zero:

$$\frac{2^3}{2^3}=2^{3-3}=2^0,\text{and}\frac{8}{8}=1$$

So $2^0=1$, and the same argument works for every non-zero base. This page covers everything in the ICSE Class 7 Mathematics chapter on exponents: exponential form and prime factors, powers of integers and fractions, the laws of exponents, and negative indices.

Exponential form writes a repeated multiplication as $a^m$, where $a$ is the base and $m$ is the index (or exponent) — 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, the index is 5, and $2^5$ is read as 2 raised to the power 5. Note that $2^1=2$, and an index of 1 is usually not written.

To write a number as a product of prime factors in exponential form, factorise it and then group the repeats. For 360:

$$360=2\times 2\times 2\times 3\times 3\times 5=2^3\times 3^2\times 5$$

Another: $1080=2^3\times 3^3\times 5$.

Large numbers become manageable this way — a lakh is $10^5$ and a crore is $10^7$, which is exactly why the form is used.

The confusion to clear up is between the base and the index. $2^3=8$ but $3^2=9$ — swapping them gives a different number, so $a^m$ and $m^a$ are not the same thing.

Look at whether the index is even or odd. An even index gives a positive result, an odd index a negative one.

$$(-2{)}^3=(-2)\times (-2)\times (-2)=-8$$

$$(-2{)}^4=(-2)\times (-2)\times (-2)\times (-2)=+16$$

The reason is the sign rule from integers: each pair of negative factors gives a positive, so an even count pairs off completely while an odd count leaves one negative over.

For fractions, the index applies to the numerator and the denominator alike:

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

The brackets are what decide the answer, and leaving them out changes it. $(-2{)}^4=16$, because the whole of $-2$ is raised to the power 4. But $-2^4=-16$, because there only the 2 is raised to the power and the minus sign stays outside. Always write the bracket when the base is negative.

A negative index means take the reciprocal and make the index positive:

$$a^{-m}=\frac{1}{a^m}$$

Worked examples:

$$2^{-3}=\frac{1}{2^3}=\frac{1}{8}$$

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

$$\frac{1}{5^{-2}}=5^2=25$$

The reason follows from the division law. Since $\frac{2^3}{2^5}=2^{-2}$, and also $\frac{8}{32}=\frac{1}{4}=\frac{1}{2^2}$, the two must be equal.

Comparing two powers usually means evaluating them, because the larger base does not always win:

$$2^5=32\text{and}{5}^2=25$$

so $2^5>5^2$, even though 5 is the bigger base. Likewise $2^{10}=1024$ is far larger than $10^2=100$.

The misconception worth naming is that a negative index makes the number negative. It does not — it makes it a fraction. $2^{-3}=\frac{1}{8}$ is positive, and only a negative base with an odd index produces a negative answer.