Prealgebra · Unit 10 · Topic 10.4

Exponent Rules and Scientific Notation

Derive product, quotient, zero, and negative exponent rules; read a powers graph; and convert and calculate with scientific notation.

What the exponent counts

The expression \(2^4\) means \(2\cdot2\cdot2\cdot2=16\), not \(2\cdot4\). The base is 2 and the exponent counts four factors. This lets us describe a value without writing a long product. For a positive whole-number exponent \(n\), \(a^n\) is a product of \(n\) copies of the base \(a\).

The graph records \(2^n\) for whole-number exponents from 0 through 4. Each step right doubles the preceding value: \(1,2,4,8,16\). The vertical axis is a linear value scale, so the height difference grows: the last jump is \(16-8=8\), much larger than the first jump \(2-1=1\). Exponents describe repeated multiplication, not a fixed amount of addition.

Original bar graph of y equals two to the n for integer n from zero through four. Bar heights are one, two, four, eight, and sixteen on a vertical value axis from zero to sixteen. Each bar is twice the previous bar. — Original graph of exact mathematical values, not measured data. Only the five integer exponents shown are plotted; each increase of 1 in \(n\) doubles \(2^n\).

Combine powers by counting factors

When two powers have the same base, their factors join into one longer product. For example,

\[ 2^3\cdot2^4 =(2\cdot2\cdot2)(2\cdot2\cdot2\cdot2) =2^{3+4} =2^7 =128. \]

For any base \(a\) and nonnegative whole numbers \(m,n\), this gives \(a^m a^n=a^{m+n}\) wherever the powers are defined. The exponent counts add because the factors are joined. By contrast, \(2^3+2^4=8+16=24\), not \(2^7\): addition does not join factors.

Division removes matching factors, provided the base is nonzero. Cancel three factors in \(5^6/5^3\), leaving three:

\[ \frac{5^6}{5^3}=5^{6-3}=5^3=125. \]

More generally, for \(a\ne0\), \(\frac{a^m}{a^n}=a^{m-n}\). Why require \(a\ne0\)? The original denominator \(a^n\) cannot be zero. The same rule explains the zero exponent: since \(\frac{a^3}{a^3}=1\), it must also equal \(a^{3-3}=a^0\). Hence \(a^0=1\) for \(a\ne0\); it is not zero. We do not use this argument to assign a value to \(0^0\).

If more factors are removed than remain, the exponent becomes negative. Check \(10^2/10^5\) two ways:

\[ \frac{10^2}{10^5} =\frac{1}{10^3} =10^{2-5} =10^{-3} =0.001. \]

Thus \(a^{-n}=1/a^n\) for \(a\ne0\) and positive integer \(n\). A negative exponent does not make a positive base negative: \(10^{-3}\) is positive. And a power raised to another power repeats a whole group of factors, so \((a^m)^n=a^{mn}\); for example, \((2^3)^2=2^3\cdot2^3=2^6=64\). These rules rely on the indicated operations—multiplication, division, or a power of a power—not on similar-looking addition.

Write very large or small numbers with powers of ten

Scientific notation expresses a nonzero number as \(c\times10^n\), where \(n\) is an integer and \(1\leq |c|<10\). For a positive number, this is the familiar \(1\leq c<10\); a negative number keeps its minus sign in \(c\). This range makes the notation standardized: \(72\times10^4\) has the right value but is not normalized, while \(7.2\times10^5\) is.

To rewrite \(720{,}000\), place the decimal after its first nonzero digit: \(7.2\). We must multiply \(7.2\) by \(100{,}000=10^5\) to recover the original value:

\[ 720{,}000=7.2\times10^5. \]

For the small number \(0.00072\), the coefficient is again \(7.2\), but now it must be divided by \(10{,}000\). Since \(10^{-4}=1/10^4=0.0001\),

\[ 0.00072=7.2\times10^{-4} =7.2\times0.0001. \]

The sign of the exponent tells us whether the power of ten enlarges or shrinks the coefficient. A useful check is to multiply back: \(7.2\times10^{-4}\) must be much smaller than 7.2. For a nonzero number already between 1 and 10, the exponent is zero, such as \(6.4=6.4\times10^0\). Zero itself is simply 0; it has no normalized scientific-notation coefficient satisfying \(1\leq|c|<10\).

Powers of ten also simplify calculations, but both the coefficients and powers must be handled:

\[ (3\times10^4)(2\times10^{-2}) =(3\cdot2)\times10^{4+(-2)} =6\times10^2 =600. \]

We multiplied the coefficients \(3\) and \(2\), then added exponents only for the two factors with base 10. As a magnitude check, \(3\times10^4=30{,}000\) and \(2\times10^{-2}=0.02\); their product is indeed 600. When a coefficient falls outside the interval from 1 to 10 after multiplication, adjust it and the exponent together: \(12\times10^3=1.2\times10^4\), because dividing 12 by 10 must be balanced by multiplying \(10^3\) by 10.