2.2 Exponential Notation and Operations

You first brushed against this last year, with squares and cubes:

n \times n = n^2 \ (\text{"}n\text{ squared"}), \qquad n \times n \times n = n^3 \ (\text{"}n\text{ cubed"}).

We just keep climbing:

n \times n \times n \times n = n^4, \qquad \underbrace{n \times n \times \cdots \times n}_{7 \text{ times}} = n^7.

The Key Idea In general, n^a stands for n multiplied by itself a times. Here n is the base and a is the exponent (also called the power or index). For instance, 5^4 = 5 \times 5 \times 5 \times 5 = 625. We read this “5 raised to the power 4.” The list 5^1, 5^2, 5^3, 5^4, \dots are the powers of 5. So 5^4 is the exponential form of 625.

Guard against muddling multiplication with addition: 4 + 4 + 4 = 3 \times 4 = 12, whereas 4 \times 4 \times 4 = 4^3 = 64. Repeated addition lands you on a product; repeated multiplication lands you on a power.

Powers cope happily with negative bases and with letters:

(-4)^3 = (-4)\times(-4)\times(-4) = -64, \qquad a\times a\times a\times b\times b = a^3 b^2.

Worked Example

Write 43200 as a product of prime factors in exponential form.

Keep dividing by primes: 43200 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5. Gathering equal primes, 43200 = 2^6 \times 3^3 \times 5^2.

Worked Example

Evaluate 6^2 \times 2^3 and 5 \times 3^4.

Compute each power, then multiply: 6^2 \times 2^3 = 36 \times 8 = 288, \qquad 5 \times 3^4 = 5 \times 81 = 405. Notice that the exponent attaches only to the number directly beneath it, in 5 \times 3^4 the 5 is not raised to the 4th power.

Math Talk Is (-1)^7 positive or negative? And (-1)^{40}? (A negative base under an odd power stays negative; under an even power it flips positive. So (-1)^7 = -1 and (-1)^{40} = +1.) Check whether (-3)^2 = 9. And what should we make of 0^2, 0^4, or 0^n in general?

Figure it Out: Exponential Form

Practice

Express in exponential form: (i) 9\times9\times9\times9 (ii) t\times t (iii) k\times k\times k (iv) 4\times4\times6\times6\times6 (v) 3\times3\times m\times m (vi) x\times x\times y\times y\times y\times y\times z
Express each as a product of powers of its prime factors in exponential form: (i) 600 (ii) 756 (iii) 880 (iv) 4500
Write the numerical value of each: (i) 4\times 10^3 (ii) 6^2\times 2^3 (iii) 5\times 3^4 (iv) (-2)^2\times(-4)^2 (v) 2^2\times 10^5 (vi) (-3)^3\times(-10)^4

Show solutions

1. 9^4 (ii) t^2 (iii) k^3 (iv) 4^2\times 6^3 (v) 3^2\times m^2 (vi) x^2\times y^4\times z.
1. 600 = 2^3\times 3\times 5^2 (ii) 756 = 2^2\times 3^3\times 7 (iii) 880 = 2^4\times 5\times 11 (iv) 4500 = 2^2\times 3^2\times 5^3.
1. 4000 (ii) 36\times 8 = 288 (iii) 5\times 81 = 405 (iv) 4\times 16 = 64 (v) 4\times 100000 = 400000 (vi) (-27)\times 10000 = -270000.

The lantern-maker’s workshop: multiplying powers

A lantern-maker keeps four shelves in her workshop. Each shelf holds four crates; each crate stores four boxes; each box guards four lanterns; each lantern carries four candles; and each candle is wrapped in four sheets of coloured paper. How many sheets of paper does the workshop hold?

Count tier by tier: 4 shelves, then 4 \times 4 = 16 crates, then 16 \times 4 = 64 boxes, then 64 \times 4 = 256 lanterns, and so on. Each tier multiplies by 4 again. Tallying all six tiers of “fours”:

4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^6.

Here is the slick part. We already know 4^4 = 256 (the lanterns). To climb to 4^6 we need only two more 4s, that is 4^2 = 16:

4^6 = \underbrace{(4\times4\times4\times4)}_{4^4} \times \underbrace{(4\times4)}_{4^2} = 256 \times 16 = 4096.

So 4^6 = 4^4 \times 4^2. We could just as well have split it as 4^3 \times 4^3, the exponents always add to 6.

Law of Exponents: Multiplication Multiplying two powers that share the same base, you add the exponents: n^a \times n^b = n^{a+b}, \qquad \text{where } a, b \text{ are counting numbers.} For instance, q^5 \times q^4 = (q\cdots q)\times(q\cdots q) = q^{9}.

Raising a power to a power

Take 6^6. Its six 6s can be bundled in different ways:

6^6 = (6\times6\times6)\times(6\times6\times6) = 6^3 \times 6^3 = (6^3)^2, 6^6 = (6\times6)\times(6\times6)\times(6\times6) = 6^2\times6^2\times6^2 = (6^2)^3.

So (6^3)^2 = (6^2)^3 = 6^6. The two exponents got multiplied: 3\times 2 = 6 = 2\times 3. (Check: (6^3)^2 = 216^2 = 46656 = 6^6.)

Law of Exponents: Power of a Power \left(n^a\right)^b = \left(n^b\right)^a = n^{a\times b} = n^{ab}, \qquad a, b \text{ counting numbers.} For example 2^{12} = (2^3)^4 = (2^4)^3, and indeed (2^4)^3 = 16^3 = 4096 = 2^{12}.

The water tanks and combining bases

In a garden tank the number of water hyacinths doubles every day, and after 30 days the surface is fully blanketed. On which day was it half covered? Because every day doubles the count, the day before full is the half-full day, the 29th! In exponential form the count reads 2^{30} when full and 2^{29} when half full.

Now a second tank triples its plants daily. Imran floats a single hyacinth in the doubling tank; after 3 days it holds 2^3 plants. He scoops them all into the tripling tank, where for 3 further days they triple, giving 2^3 \times 3^3. Had he reversed the order? Then 1 \times 3^3 \times 2^3, the very same total. Regroup it:

3^3 \times 2^3 = (3\times2)\times(3\times2)\times(3\times2) = (3\times2)^3 = 6^3 = 216.

Law of Exponents: Same Exponent m^a \times n^a = (m\times n)^a, \qquad \text{and likewise } \frac{n^a}{m^a} = \left(\frac{n}{m}\right)^a. For example 5^3 \times 2^3 = (5\times2)^3 = 10^3 = 1000, and \dfrac{12^4}{4^4} = \left(\dfrac{12}{4}\right)^4 = 3^4 = 81.

Counting the choices

Priya owns 5 kurtas and 4 scarves. For each scarf she may pair any of the 5 kurtas, making 5\times 4 = 20 outfits. When choices stack one after another, we multiply the options at each stage.

This is exactly how locks work. A 2-digit dial has 10\times 10 = 100 settings; a 3-digit dial has 10\times10\times10 = 1000; and a 4-digit dial has

10\times10\times10\times10 = 10^4 = 10{,}000 \text{ settings.}

If a lock instead held 5 slots, each a letter A–Z (26 choices), the number of passcodes would balloon to 26^5 = 1{,}18{,}81{,}376, a far sturdier lock!

Try This Naveen has 6 shirts, 3 caps and 2 pairs of shoes, that is 6\times3\times2 = 36 ways to get dressed. Now think about everyday codes built the same way: PIN codes (the PIN code of Warangal, Telangana is 506002), mobile numbers, vehicle registration plates. How many of each can exist? See if you can find out how these codes are actually assigned.