79 1.5. Exponents and Scientific Notation — Mathematics for Public and Occupational Health Professionals

<!– pb_fixme –>

Exponents

Exponent review:  an or BaseExponent

Exponential notation Example
Base     Exponent
an= a ∙ a ∙ a ∙ a … a 24 = 2 ∙ 2 ∙ 2 ∙ 2  = 16
Read “a to the nth” or “the nth power of a.” Read “2 to the 4th.”

 

Properties of exponents:

Name Rule Example
Product rule a^m\;a^n=a^{m+n} 2^3\;2^2=2^{3 + 2}=2^5=32
Quotient rule \frac{a^m}{a^n}=a^{m-n} \frac{y^4}{y^2}=y^{4-2}=y^2
Power rule (a^m)^n=a^{mn}

(a^m \cdot b^n)^p=a^{mp}\;b^{np}

(\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}

 (x^3)^2=x^{3\cdot2}=x^6

(t^3 \cdot s^4)^2=t^{3 \cdot 2}\;s^{4 \cdot 2}=t^6\;s^8

(\frac{q^2}{p^4})^3=\frac{q^{2\cdot3}}{p^{4\cdot3}}=\frac{q^6}{p^{12}}
Negative exponent a-n a^{-n}=\frac{1}{a^n} 4^{-2}=\frac{1}{4^2}=\frac{1}{16}
\frac{1}{a^{-n}}=a^n \frac{1}{4^{-2}}=4^2=16
Zero exponent   a0 a^0=1 15^0=1
One exponent   a1 a^1=a 7^1=7   ,   1^{13}=1
Fractional exponent a^\frac{m}{n}=\sqrt[n]{a^m} 15^\frac{2}{3}=\sqrt[3]{15^2}

 

  • Product rule: when multiplying two powers with the same base, keep the base and add the exponents.

 

  • Quotient rule: when dividing two powers with the same base, keep the base and subtract the exponents.

This law can also show that why a0 = 1 (zero exponent a0):  \frac{a^2}{a^2}=a^{2-2}=a^0=1

 

  • Power rule: when raising an expression to a power, we multiply each exponent inside the parentheses by the power outside the parentheses.

 

  • Negative exponent: a negative exponent is the reciprocal of the number with a positive exponent.

 

  • Fractional exponent: a fractional exponent is a different way of writing a radical (i.e. root) sign. The base is first taken to the exponent of m, then the nth root is found to obtain the power.

 

Simplify (do not leave negative exponents in the answer).

1) {\bf (-4)^1}=-4 a^1=a
2) {\bf (-2345)^0}=1 a^0=1
3) {\bf x^2x^3}=x^{2+3}=x^5 a^m\;a^n=a^{m+n}
4) {\bf \frac{y^6}{y^4}}=y^{6-4}=y^2 \frac{a^m}{a^n}=a^{m-n}
5) {\bf (x^4)^{-3}}=x^{4(-3)}=x^{-12}=\frac{1}{x^{12}} (a^m)^n=a^{mn}  ,  \frac{1}{a^{-n}}=a^n
6) {\bf 7b^{-1}}=7\cdot \frac{1}{b^1}=\frac{7}{b} a^{-n}=\frac{1}{a^n}  ,  a^1=a
7) {\bf (2t^3\cdot w^2)^4}=2^4 t^{3\cdot4}\cdot w^{2\cdot4}=16t^{12} w^8 (a^m \cdot b^n)^p=a^{mp}\;b^{np}
8) {\bf \frac{1}{3^{-2}}}=3^2=9 \frac{1}{a^{-n}}=a^n
9) {\bf \frac{7x^4y^{-5}}{9^0\cdot x^2y^3}}=\frac{7x^{4-2}y^{-5-3}}{1}=7x^2y^{-8}=\frac{7x^2}{y^8} a^0=1  ,  \frac{a^m}{a^n}=a^{m-n}  ,  a^{-n}=\frac{1}{a^n}
10) {\bf (\frac{e^{-3}f^2}{g^{-2}})^{-2}}=\frac{e^{(-3)(-2)}f^{2(-2)}}{g^{(-2)(-2)}}=\frac{e^6f^{-4}}{g^4}=\frac{e^6}{g^4f^4} (\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}  ,  \frac{1}{a^{-n}}=a^n

 

Simplify.

1) \bf (3x^3y^2)^2 (2x^{-3}y^{-1})^3 (-248z^{-19})^0
=3^2x^{3\cdot2}y^{2\cdot2} \cdot 2^3x^{-3\cdot3} \cdot y^{-1\cdot3}\cdot1 Remove brackets. (\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}} , a^0=1
=(3^2\cdot2^3)(x^6x^{-9})(y^4y^{-3}) Regroup coefficients and variables.
=72x^{-3}y^1 Simplify. a^m\;a^n=a^{m+n}
=\frac{72y}{x^3} Make exponent positive. a^{-n}=\frac{1}{a^n} , a^1=a
2) \bf (\frac{(2x^4)(y^5)}{3x^3y^2})^2 (\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}
=\frac{(2x^4)^2(y^5)^2}{(3x^3y^2)^2}
=\frac{2^2x^{4\cdot2}y^{5\cdot2}}{3^2x^{3\cdot2}y^{2\cdot2}} Remove brackets. (a \cdot b)^n=a^n\;b^n
=\frac{4}{9}\cdot \frac{x^8}{x^6}\cdot \frac{y^{10}}{y^4} Regroup coefficients and variables.
=\frac{4}{9}x^2y^6 Simplify. \frac{a^m}{a^n}=a^{m-n}

 

Evaluate for   a = 2,   b = 1,   c = -1.

1) {\bf (-29a^{-5}b^4c^{-7})^0}=1 a^0=1
2) {\bf (\frac{a}{b})^{-4}}=(\frac{2}{1})^{-4} Substitute 2 for a and 1 for b,
=\frac{2^{-4}}{1^{-4}}=\frac{1^4}{2^4}=\frac{1}{16} \frac{a^m}{a^n}=a^{m-n}  ,  a^{-n}=\frac{1}{a^n}  ,  \frac{1}{a^{-n}}=a^n
3) {\bf (a+b-c)^a}=[2+1-(-1)]^2=4^2=16 Substitute 2 for a and 1 for b, and -1 for c.

Scientific Notation

Scientific notation is a special way of concisely expressing very large and small numbers.

Example:       300,000,000 = 3 × 108 m/sec                             The speed of light.

 

Scientific notation: a product of a number between 1 and 10 and a power of 10.

Scientific notation Example
N × 10±n 1 ≤ N < 10 67504.3 = 6.75043 × 104
n – integer Standard form Scientific notation

 

Writing a number in scientific notation:

Step Example
  • Move the decimal point after the first nonzero digit.

 

Write in scientific notation.

1)     2340000 = 2340000. = 2.34× 106                                   6 places to the left,    × 10n

2)     0.000000439 = 4.39 × 10-7                                           7 places to the right,   × 10-n

 

Write in standard (or ordinary) form.

1)     6.4275 ×104 = 64275

2)    2.9 × 10-3 = 0.0029

 

Practice questions

1. Evaluate:

2. Simplify (do not leave negative exponents in the answer):

3. Write in scientific notation:

4. Write in standard (or ordinary) form:

 

<!– pb_fixme –>

<!– pb_fixme –>
<!– pb_fixme –>

License

Math Copyright © by sboschman. All Rights Reserved.

Share This Book