# 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 Do math in grouping symbols: ( ), [ ], { }, |  | Do all exponents from left to right Multiply and divide from left to right Add and subtract from left to right 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)

## Notes

Exponents

 Exponents are sometimes called powers The base of an exponent can be ANY type of number (negative, positive, fractional, decimal, irrational, etc.)

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

Examples of Exponent Laws

 Example: Division Law (Breakdown) The product law and fraction law both use the distribution of exponents   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! 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)

Square Roots

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.

## 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: $171$

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: $a^{10}$

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?

• 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: $\sqrt{10}, \sqrt{\frac{40}{4}}, \frac{\sqrt{40}}{\sqrt{4}}$