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6.1 – Expontents

Key Terms

  • Base – A number in exponential form. It appears beneath the exponent.
  • Exponent – A small raised number that tells you how many times to multiply the base number by itself.
  • Exponential Expression – An expression that involves an exponent.
    • The expression may contain variables, which may be the base or the exponent.
  • Fractional – Having to do with a fraction.
    • The fractional part of the number 3 \frac{2}{3} is \frac{2}{3}
  • Power – A small raised number that tells you how many times to multiply the base number by itself. It is also called an exponent.
  • Principal Square Root – The positive square root of a number.
    • The principal square root of b is written \sqrt{b}, or b^\frac{1}{2}.
    • If x is the principal square root of n, then x^2=n.
  • Radical – The symbol \sqrt{} is used to indicate a square root.
    • \sqrt[n]{} indicates the nth root.
    • \sqrt[2]{} is simply written as \sqrt{}.
  • Scientific Notation – A way to represent very large or very small numbers.
    • A number in scientific notation has the form N \cdot 10^m, where N is between 1 and 10 and m is an integer.

Review

Order of Operations

  • PEMDAS
    1. Do math in grouping symbols: ( ), [ ], { }, |  |
    2. Do all exponents from left to right
    3. Multiply and divide from left to right
    4. Add and subtract from left to right
PEMDAS
  • Example: PEMDAS: [8-(5-3)^2]\bullet2 \div (3-2)^4
    • Parenthesis / Grouping Symbols
      • [8-(2)^2]\bullet2 \div (1)^4
    • Exponents, then Start Over (Parenthesis)
      • [8-4]\bullet2 \div 1
      • 4\bullet2 \div 1
    • Multiplication and Division
      • 8 \div 1
    • Answer: 8

 

Perfect Squares

  • Numbers than can be divided by two identical factors
    • Ex. \sqrt{64}=8 because 8\bullet8=64
  • Did you know?!  No perfect square ends in the digits 2, 3, 7, or 8
  • Table of 20 Perfect Squares (below)
Perfect Squares

Notes

Exponents

  • Exponents are sometimes called powers
  • The base of an exponent can be ANY type of number (negative, positive, fractional, decimal, irrational, etc.)
Alg1B 6.1 - exponent diagram Alg1B 6.1 - Exponent Examples

 

Exponent Laws

  • Multiplication Law
    • Add the exponents if you are multiplying the SAME base
  • Division Law
    • Subtract the exponents if you are dividing the SAME base
  • Power Law
    • Multiply the exponents if you are raising an exponent to a power
  • Product Law
    • Distribute the exponent to the bases (the factors)
  • Quotient Law
    • Distribute the exponent to the bases in the numerator and the denominator
Alg1B-6.1-Exp-Laws-Summary-600x273

 

Examples of Exponent Laws

Alg1B 6.1 - Exp Laws

  • Example: Division Law (Breakdown)

Alg1B 6.1 - Exp Division

Alg1B 6.1 - Exp Laws2

  • The product law and fraction law both use the distribution of exponents

 

Alg1B 6.1 - Exp Laws Fractions

  • Notice how fractional multiplication is like distributing an exponent!

 

Powers and Roots

  • A fractional exponent can be converted into a root
    • The denominator of the fraction becomes the root
  • A root can be converted into a fractional exponent
    • The root becomes the denominator of the fraction
  • Examples
    • Power to Root: x^{\frac{1}{2}}=\sqrt[2]{x}=\sqrt{x}
    • Root to Power:  \sqrt[3]{x}=x^{\frac{1}{3}}

 

Exponent Rules for Negatives and Zeros

  • Negatives: change places (numerator or denominator) and make the power positive
  • Zeros: a base to any zero exponent equals 1, always!
Alg1B 6.1 - Exp Rules
  • Examples of Negative and Zero Powers
    • 3^{-6}=\frac{1}{3^{6}}=\frac{1}{729}
    • \frac{1}{3^{-6}}=3^{6}=729
    • \frac{4a}{3^{-6}}=4\bullet3^{6}=4a\bullet729=2916a
    • 84^{0}=1
    • -34^{0}=1

 

Principal Square Root Rules

  • Principal Square Root: b^{\frac{1}{2}}
  • For any non-negative number b, it is the number b^{\frac{1}{2}}.
  • Every positive number has two square roots: the principal square root and the opposite of the principal square root.
    • Principal square roots are ALWAYS positive
      • The principal square root of 49 is 7 (positive 7)
    • Opposites of principal square roots are negative
      • The opposite of the principal square root of 49 is -7 (negative 7)
Alg1B 6.1 - Principal Square Roots

 

Square Roots

Alg1B 6.1 - Radicals Alg1B 6.1 - Roots Powers
Alg1B 6.1 - Root Examples

 

Scientific Notation

  • It provides a simple way of writing very large or small numbers, such as 4,334,566,000,000,000,000,000,000,000,000!
    • 4.334566\bullet10^{30}
  • Steps for Solving Scientific Notation
    • How to Write Large Numbers in Scientific Notation
      • Step 1: Move the decimal point to the left until it’s between the ones and tenths.
      • Step 2: Use that decimal number for N.
      • Step 3: Count the number of places you moved the decimal point.
      • Step 4: Use that number for positive m.
    • How to Write Small Numbers in Scientific Notation
      • Step 1: Move the decimal point to the right until it’s between the ones and tenths.
      • Step 2: Use that decimal number for N.
      • Step 3: Count the number of places you moved the decimal point.
      • Step 4: Use that number for negative m.
Alg1B 6.1 - Scientific Notation Alg1B 6.1 - SciNotation
Alg1B 6.1 - Sci Not Small
Alg1B 6.1 - SciNotation Ex

Examples

Order of Operation Example

  • Ex 1. Evaluate: 4(2+5)^2-5^2
    P: 4(7)^2-5^2
    E: 4(49)-25
    MD: 196-25
    AS: 171
  • Answer: 171

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Exponent Examples

  • Ex 2. Which of the following is equal to the expression below?
    • 2^{-3}
    • Setup: \frac{1}{2^3}
      • Answer: \frac{1}{8}

 

  • Ex 3. Which of the following is equal to the expression below?
    • Setup: (6^{-5})^2
    • Simplify: (\frac{1}{6^{10}})
      • Answer: (\frac{1}{6})^{10}

 

  • Ex 4. Which expression is equal to the expression below?
    • (6 \cdot 9)^{5}
    • The exponent can be distributed
      • Answer: 6^{5} \cdot 9^{5}

 

  • Ex 5. Which of the following is equal to the fraction below?
    • (\frac{4}{5})^{6}
    • Distribute the negative sign
      • Answer: \frac{4^{6}}{5^{6}}

 

  • Ex 6. Evaluate: 3^{0}
    • Anytime an exponent is raised to the zero power, it equals 1
      • Answer: 3^{0}=1

 

  • Ex 7. What is the equivalent of: (a^{4})(a^{5})
    • Add exponents when you are multiplying with the same base
      • Answer: (a)^{9}

 

  • Ex 8. What is the equivalent of: \frac{a^{15}}{a^{5}}
    • Subtract exponents when you are dividing with the same base
      • Answer: a^{10}

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Square Root Examples

  • Ex 9. What is the value of the exponential expression below?
    • 16^{\frac{1}{2}}
    • Any exponential fraction should be translated to a radical. In this case: \sqrt[2]{}.
    • Setup: \sqrt[2]{16}, which can be simplified to \sqrt{16}.
      • Answer: 4

 

  • Ex 10. If z is the principal square root of 8, what must be true?
    • Answer: z=8^{2}, and z is positive.

 

  • Ex 11. Simplify this radical expression.
    • \sqrt{4w}
    • Since only 4 can be squared, you must leave w inside the root.
      • Answer: 2 \sqrt{w}

 

  • Ex 12. Simplify this radical expression.
    • \sqrt{\frac{q}{36}}
    • Since only 36 can be squared, you must leave q inside the root.
      • Answer: \frac{\sqrt{q}}{6}

 

  • Ex 13. Simplify: \sqrt{28}
    • Breakdown into factors, looking for perfect squares: \sqrt{4 \cdot 7}
    • Since 4 is a perfect square, take the square root of 4 and leave the 7 inside.
      • Answer: 2 \sqrt{7}

 

  • Ex 14. Simplify: (\frac{4}{9})^{\frac{1}{2}}
    • Convert the fraction to a square root: \sqrt{\frac{4}{9}}
    • Break the square root down into individual square roots: \frac{\sqrt{4}}{\sqrt{9}}
    • Since 4 and 9 are both perfect squares, take the square root of each:
      • Answer: \frac{2}{3}

 

  • Ex 15. What numbers are square roots of 64?
    • Answer: +8 and -8

 

  • Ex 16. What numbers are square roots of 81?
    • Answers: +9, -9, -(81)^{\frac{1}{2}}, (81)^{\frac{1}{2}}

 

  • Ex 17. Which of the following are equal to the expression below?
    • 2 \sqrt{49}
      • Answers: +14, -14, -(196)^{\frac{1}{2}}, (196)^{\frac{1}{2}}, \sqrt{196}

 

  • Ex 18. Which of the following are equal to the expression below?
    • \frac{\sqrt{40}}{2}
      • Answer: \sqrt{10}, \sqrt{\frac{40}{4}}, \frac{\sqrt{40}}{\sqrt{4}}

 


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