Negative Exponents

In math, exponents define the number of times a number is multiplied by itself. They are written with a base number like this: $4^3$

where 4 is the base number and 3 is the exponent. The previous term is read as "4 to the 3rd power" and indicates that 4 is multiplied by itself 3 times, or $4\times 4\times 4$

An exponent can be positive, negative, or zero, but they are handled differently.

We know that a positive exponent tells us to multiply a number by itself a particular number of times. A negative exponent, on the other hand, tells us how many times to divide the base number. In practice, the negative exponent means that we have to multiply the reciprocal of the base a certain number of times.

An example of a negative exponent is $4^{-2}$

We can write $4^{-2}$ as ${\frac{1}{4}}^2$

The value of $4^{-2}$ is therefore $\frac{1}{16}$ because $\frac{1}{4}\times \frac{1}{4}=\frac{1}{16}$

There are two important rules of negative exponents that simplify solving problems involving them.

For every number "a" with a negative exponent, "-n", multiply the value of the reciprocal of the base number according to the value of the exponent number.

For example, $6^{-3}$ . The base number is 6 and the exponent is -3.

According to the rule, $6^{-3}$ is written as $\frac{1}{6^3}$ , which is written as " $\frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}$ ", which equals $\frac{1}{216}$ .

This means the value of $6^{-3}=\frac{1}{216}$ .

For every number "a" in the denominator with a negative exponent "-n", the result can be written in the form of $a\times a\times a$ .. according to the exponent.

As an example, take the fraction $\frac{1}{3^{-3}}$ .

This time, the negative exponent is in the denominator of the fraction.

Here, $\frac{1}{3^{-3}}$ can be rewritten as , which is equal to $3\times 3\times 3$ , which equals 27.

Therefore, $\frac{1}{3^{-3}}=27$ .

You can use the product of powers property to show how negative exponents work as follows.

Here's the plain text representation of the given equations:

$7^{-2}\times 7^2=7^{-2+2}=7^0$

We know that $7^2$ equals 49, and we know that $7^0$ equals 1. This tells us that $7^{-2}\times 49$ equals 1.

What number times 49 equals 1? That would be its multiplicative inverse, $\frac{1}{49}$ .

Therefore, $7^{-2}$ equals $\frac{1}{49}$ .

The general rule for this is written as:

For all real numbers a and b where $a\ne 0$

$a^{-b}$ is defined as $\frac{1}{a^b}$ ,

$\frac{1}{a^{-b}}$ is defined as $a^b$ .

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