Just as **multiplication** corresponds to **repeated addition**,

**Exponentiation** corresponds to **repeated multiplication**.

## Exponentiation: Base and Exponent

In the above-generalized formula for exponentiation, **x** is called the **base**, and the **superscript** (raised letter) **n** is called the **exponent**.

x^{n} is read as “x raised to the nth power” which is often abbreviated as “x to the nth”.

## Exponentiation: Exceptional cases

When the exponent is 2, it is considered a special case so that x^{2} is commonly referred to as x squared (instead of x to the 2nd power). The fact is, the area of a square with side x is x^{2}, and thus the origin of the term “x squared.”

Similarly, the exponent 3 is also considered as a special case as x^{3} is commonly referred to as “x cubed” (instead of x to the 3rd

power). The volume of a cube with side x is x^{3} and thus the origin of the term “x cubed”

## Few more examples of Exponentiation with a power of 10

Let’s see few more examples based on what we learned so far

If the number 10 appears 4 times in repeated multiplication, we can write 10•10•10•10 which equals 10000. This expression is also written as a power, 10^{4}, where 10 is the base and 4 is the exponent.

### Expressing a number in a power of 10

Any number can be expressed in terms of powers of 10. For example,32,586.49 is also written as 32,586.49_{10}―where the 10 subscript designates the base of the number). By default the base of any number is always considered to be 10. It can be expressed as :

3•10^{4} + 2•10^{3} +5•10^{2} + 8•10^{1} + 6•10^{0} + 4•10^{-1} + 9•10^{-2}.

We automatically assume the base 10 if no other base is specified.

### Expressing a number in a power of 2

Next, consider a number with base 2. For example 111101_{2} (note that the 2 subscript to the right of the number designates the base of the number. 2 indicates the number is base 2 or binary). You can express this binary number in terms of powers of 2, or

1•2^{5} + 1•2^{4} +1•2^{3} + 1•2^{2} + 0•2^{1} + 1•2^{0} = 32 + 16 + 8 + 4 + 0 + 1 = 6110.

We can also use exponents to simplify the repeated product of the variable x:

x•x•x•x = x^{4}

### Negative Exponents:

Exponents may also be negative. By definition, x^{–1} = 1/𝑥, so that 2^{–1} = 1/2. In general, any base raised to a negative exponent, x^{–n} can be rewritten with the same base having a positive exponent as 1/𝑥^{𝑛}. So that we have

x^{-n} =1/𝑥^{n}

Thus, 10^{-4} is equivalent to:

10^{-4} = 1/10^{4} =1/10•10•10•10 =1/10000= 0.0001

Similarly, if we have a **negative exponent in the denominator,** 1/𝑥^{−𝑛}, we can write the same base with a positive exponent in the numerator; thus,

1/𝑥^{−𝑛} = x^{n}

#### Example of negative exponents:

As another example, we can write positive exponents as negative exponents and negative exponents in the form of positive exponents.

a^{−4}b^{2}c^{−3}/d^{−5} in terms of only positive exponents:b^{2}d^{5}/a^{4}c^{3}

Any number raised to power 0 is always 1

Any number raised to power 1 is always that number.

## Few Simplified properties of Exponentiation to remember:

Since 3^{2}•3^{4} = (3•3)(3•3•3•3) = 3^{6}, this leads to the property that

x^{a} • x^{b}= x^{a + b}.

3^{4}/3^{2} is equivalent to3^{4}/ 3^{2} = 3•3•3•3•3 / 3•3 = 3^{2}. This leads to the property that

𝑥^{𝑎}/𝑥^{𝑏} = x^{a – b}.

Finally, we have the case of (3^{2})^{3} which equals: (3 • 3)^{ 3} = (3 • 3) (3 • 3) (3 • 3) = 3^{6}. This leads to the property that

(x^{a})b= x^{a•b}.

Now, let’s consider a combination of these exponentiation properties by considering: (4x^{-3}y^{-4})^{-2}.

This is equal to 4^{-2}(x^{-3})^{-2}(y^{-4})^{-2} = (1/4^{2}).x^{6}y^{8} = 𝑥^{6}𝑦^{8}/16 .Alternatively,

(4x^{-3}y^{-4})^{-2} = 1/(4𝑥^{−3}𝑦^{−4})^{2} = 1/4^{2}𝑥^{−3•2}𝑦^{−4•2} = 1/16𝑥^{−6}𝑦^{−8} =𝑥^{6}𝑦^{8}/16 .

**Examples are taken from math book with solutions provided by hackonmath**

## Conclusion of Exponentiation:

- x
^{0}= 1 - x
^{1}= x - x
^{n = 1/x-n} - 1/x
^{-n = xn} - x
^{-1}= 1/x - x
^{a}+x^{b}= x^{a+b} - x
^{a}/x^{b}= x^{a-b} - (x
^{a})^{b}= x^{a.b}

`

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