Exponents & Roots (Strategy Guide III)

Question 1

Rule 1: Adding and Subtracting Exponents

Answer 1

May only add or subtract LIKE TERMS
(i.e. same base & exponent)

CANNOT add or subtract if either base or exponents are different (ie: must FACTOR)

e.g.) 2^29 - 2^28 = 2^28(2-1) = 2^28(1) = 2^28

Question 2

Rule 2: Multiplying Exponents

Same Base
Same Exponent
Same Base & Exponent

Answer 2

Same Base = add EXPONENTS

Same Exponent = multiply BASE

Same Base & Exponent = Either/OR of above
(e.g. 3^2 x 3^2 = 3^4 OR 9^2)

Question 3

Rule 3: Dividing Exponents

Answer 3

Same Base = SUBTRACT exponents

Same Exponent = DIVIDE base

Same Base & Exponent = quotient of coefficients times 1
eg [4^5 / 4^5 = 1]. … eg [28y^5 / 7y^5 = 4]

Question 4

Rule 4: Negative Exponents

Answer 4

FLIP the BASE and make the exponent positive

x^-2 = 1/x^2 … 1/2x^-4 = x^4/2

Question 5

Rule 5: Powers of One and Zero

Answer 5

Power of One: Any term to the first is equal to itself

Power of Zero: Any term to the power of zero has a value of 1 (except for zero)!

Question 6

Rule 6: Resolving Parentheses…Simple vs Complex Expression

Answer 6

1.) Determine whether it’s simple of complex expression

Simple(no addition or subtraction) = distribute exponent to each term within parentheses

Complex (contain add or sub) = combine terms (within) and distribute within.

Question 7

Rule 7: Consecutive Exponents

expressions with exponents INSIDE and OUTSIDE parentheses

Answer 7

Simple Expression: Distribute outside exponent to EVERY TERM within.

Complex: Combine exponent within parentheses before multiplying exponents with EVERY term within.

Question 8

Rule 8: Fractional Exponents (eg 64^1/3)

Answer 8

Bottom = root of base

Top = Power new base should be raised to.

Question 9

Recap:

1. Addition or Subtraction
2. Multiply (Same Exponent/Base/Both)
3. Divide (Same Exponent/Base/Both)
4. Negative Exponents
5. Powers of One and Zero
6. Resolving Parentheses
7. Consecutive Exponents
8. Fraction Exponents

Answer 9

1. Like terms only
   2a. Same Base = add exponents
   2b. Same Exponent = multiply base
   2c. Same Base & Exponent = Either/Or
   3a. Same Base = subtract exponents
   3b. Same Exponent = divide base
   3c. = quotient times 1
2. Flip BASE, make positive
3. One = itself; Zero = Value or 1 (except 0)
4. Simple (distribute) vs Complex (combine then distribute)
5. Simple (distribute) vs Complex (combine)
6. Bottom = root / Top = raised power

Question 10

Exponential Equations (exponents on both sides of equation)

Inequalities with exponents = same order of operations

Answer 10

• Rephrase so Eq. has same base OR same exponent
• Next, eliminate whatever is the same
• Solve for what’s left
• If unsure, breakdown the base is best 1st move.
• If exponents are already the same…ELIMINATE and solve for what’s left.
• Exponents Eqs with addition or subtraction; it can be helpful to rewrite the term via reversing the rule.

Question 11

Simplifying Roots: (must always be in their simplest form)

– Simple Roots (eg: √16 / √9 = 4/3)

– Complex Roots
(eg: √36 + √36 = √72 = √36 x √2 = 6√2)

Answer 11

Simple = no add or subtract…break it up

Complex = contains add or subtract…combine, then break

• SIMPLIFY BEFORE ADDING OR SUBTRACTING*
  eg: √20 + √45 = 2√5 + 3√5 = 5√5

Question 12

Order Of Operations

• Same vs different radical
• Multiply
• Divide

Answer 12

Operations:

– Same Radical...add coefficients
– Different Radical...no further simplification 
– Multiply...combine what’s common 
– Divide...break out, cross cancel
(eg √27 / √7 = √7 x √4 / √7  = √4 = 2)

Question 13

Order of Ops (Cont’d)

• Radicals in Denominator
• Approximating Roots if root has coefficient

Answer 13

– Radicals in Denominator…break down..multiply both sides by remaining radical
(eg 12√10 / 3√15 = 4√2 /√3 = 4√2 x √3 / √3 x √3 =
4√6 / 3)

– Approx Roots –> If root has coefficient, convert coeff into a root (eg 3√7 = √9 x √7 = √63 ≈ √64 ≈ 7.9

Question 14

What is a conjugate?

• Complex Denominators w/ radicals

Answer 14

Conjugate = switching addition / subtraction sign

Complex Denoms with radicals are simplified with conjugate
eg: 7 / 3 + √8 = 7/3+√8 x 3 - √8 / 3 - √8

Question 15

Roots as Exponents

What does the root become?

Answer 15

ROOT = Denominator

Question 16

Complex Square Roots …how do you simplify

Answer 16

FACTORING

eg: √50(29) + 10(15) = √5x5x2(29) + √5x5x3x2 =
√25(58) + √25(6) = √25(58+6) = 5 x 8 = 40

Question 17

Roots & Decimals…how do you find number of decimal places?

Answer 17

Decimal Places of Answer x “Root Number” = Decimal places of Original