Every Number Can Be Written as One Digit and a Power of Ten

As one digit, a decimal part, and a power of ten. So $5600000$ becomes $5.6\times 10^6$, and $0.00042$ becomes $4.2\times 10^{-4}$.

The power of ten does the counting for you, which is why this form is used for anything very large or very small. This page covers everything in the CBSE Class 8 Mathematics chapter's second part: zero and negative exponents, rewriting with positive exponents, scientific notation, and comparing extreme quantities.

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

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

So $2^0=1$. The same argument works for every base except zero:

$$n^0=1(n\ne 0)$$

Examples:

$$5^0=1,(-7{)}^0=1,{\left(\frac{3}{4}\right)}^0=1,(100{)}^0=1$$

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

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

Examples:

$$2^{-3}=\frac{1}{2^3}=\frac{1}{8},10^{-2}=\frac{1}{100}=0.01,5^{-1}=\frac{1}{5}$$

This also 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 misconception worth naming is that a negative exponent 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 exponent gives a negative answer.

Move each factor across the fraction bar, flipping the sign of its exponent as it goes.

Worked examples.

$$3^{-2}=\frac{1}{3^2}=\frac{1}{9}$$

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

For a fractional base, a negative exponent simply turns the fraction upside down:

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

The same laws as before still apply, with negative numbers in the arithmetic:

$$3^{-2}\times 3^5=3^{-2+5}=3^3=27$$

$$\frac{2^{-3}}{2^{-5}}=2^{-3-(-5)}=2^2=4$$

$$(4^{-2}{)}^3=4^{-6}=\frac{1}{4^6}$$

A mixed simplification:

$$2^{-2}\times 2^4\times 2^{-1}=2^{-2+4-1}=2^1=2$$

The habit that prevents errors is to handle the exponent arithmetic first, keeping the signs, and convert to a positive exponent only at the end. Trying to flip each factor as you go usually costs a sign — and note that subtracting a negative exponent adds, which is why $-3-(-5)$ came out as $+2$ above.

Put both in scientific notation and compare the exponents first. Only if the exponents are equal do you compare the leading numbers.

Worked examples.

Compare $3.2\times 10^8$ and $7.5\times 10^7$. Since $8>7$:

$$3.2\times 10^8>7.5\times 10^7$$

The larger exponent wins even though $7.5$ is bigger than $3.2$.

Compare $4.1\times 10^5$ and $9.3\times 10^5$. The exponents match, so compare the leading numbers:

$$9.3\times 10^5>4.1\times 10^5$$

With negative exponents, remember that a more negative exponent means a smaller number:

$$2\times 10^{-3}>8\times 10^{-5}$$

since $10^{-3}=0.001$ while $10^{-5}=0.00001$.

Ordering a set. Arrange $6.2\times 10^4$, $1.5\times 10^6$, $8.9\times 10^3$ and $2.1\times 10^6$ in increasing order. Sorting by exponent first, then by leading number:

$$8.9\times 10^3,6.2\times 10^4,1.5\times 10^6,2.1\times 10^6$$

The trap is comparing the leading digits first, and it gives the wrong answer whenever the exponents differ. A quantity written as $9.9\times 10^3$ is far smaller than one written as $1.1\times 10^6$, because the power of ten outweighs everything in front of it — which is exactly why the form is so useful for comparing things of wildly different size.