3. Exponents & Roots Flashcards
Parentheses are used to separate a coefficient from a base that contains a numeral, whereas
parentheses are not used to separate a coefficient from a base that is a variable
2y^3 vs. 3(5y)^4
8 exponent rules:
1) Addition and subtraction
2) Multiplication
3) Division
4) Negative exponents
5) Power of 0 and 1
6) Parentheses
7) Consecutive
8) Fractional
Exponent rule #1 : Addition & subtraction
Terms that contain exponents may only be added or subtracted if they are like terms.
Like terms (def)
the same base and the same exponent
2x^3 and 6x^3
Exponent rule #2: Multiplication
Terms that contain exponents may only be multiplied if they share the same base or the same exponent.
1) Same bases
Terms that share the same base can be multiplied by adding their exponents.
2) Same exponents
Terms that share the same exponent can be multiplied by multiplying their bases.
3) Same bases, same exponents
Terms that share the same base and the same exponent can be multiplied either by adding their exponents OR by multiplying their bases!
Terms that share the SAME BASE can be multiplied by ____
adding their exponents.
5^4 × 5^5 = 5^9
2x^4 × 4x^4 = 8x^8
Bases should not be multiplied, but coefficients well.
Terms that share the SAME EXPONENT can be multiplied by ______
multiplying their bases.
2^2 × 3^2 = 6^2
The coefficients of bases with similar exponents should also be
multiplied.
Terms that share the same base AND the same exponent can be multiplied either by _____ or by _______
adding their exponents
OR by multiplying their bases
3^2 ×3^2 = 3^4 or 9^2
5^x × 5^x = 5^2x or 25^x
Exponent rule #3: Division
Terms that contain exponents may only be divided if they share the same base or the same exponent.
1) Same bases
Terms that share the same base can be divided by subtracting their exponents.
2) Same exponents
Terms that share the same exponent can be divided by dividing their bases.
3) Same bases and same exponents
Terms that share the same base and the same exponent always equal the quotient of their coefficients times 1, since any term divided by itself equals 1!
Terms that share the SAME BASE can be divided by ________
subtracting their exponents.
The bases should not be divided, but the coefficients should
Terms that share the SAME EXPONENT can be divided by _______
dividing their bases.
Terms that share the same base AND the same exponent always equal _________
the quotient of their coefficients times 1, since any term divided by itself equals 1!
5^3 ÷ 5^3 = 5^3 / 5^3 = 1
Exponent rule #4: Negative exponents
Flip the base!
Any term with a negative exponent can be rewritten by flipping the base and making the exponent
positive.
If the negative exponent is contained in the denominator, flip the base into the numerator.
Any term with a negative exponent can be rewritten by ________
flipping the base and making the exponent
positive.
If the negative exponent is contained in the denominator, _________
flip the base into the numerator.
1/X^(-4) = x^4
Exponent rule #5: The powers of 1 and 0
Any term raised to the first power is known sa power of 1
Any term raised to the zero power is known as a power of 0
Any term to the zero power has a value of one, save for zero itself!
Any term to the zero power has a value of ______
one, save for zero itself!
0^0 = undefined
Any term to the first power is equal to ______
itself.
Exponent rule #6: Resolving parentheses
Before resolving the parentheses of an exponential expression, first determine whether the given term is simple or complex.
A simple expression contains no addition or subtraction within its parentheses.
A complex expression contains addition or subtraction within its parentheses.
To resolve the parentheses of a simple expression, _________
distribute the exponent outside the parentheses to each term within the parentheses.
(2x)^3 = 2^3x^3 = 8x^3
To resolve the parentheses of a complex expression, _______
combine the terms within the parentheses and then distribute the exponent
(2+3)^4=(5)^4 =625
If the terms within the parentheses cannot be combined,______
Instead, the
entire parentheses must _______
the exponent cannot be distributed.
be multiplied out.
(x +y)^2 =(x+y)(x+y)
Not (x+y)^2 = x^2 +y^2
Exponent rule #7: Consecutive exponents
Any expression that contains exponents inside and outside its parentheses can be rewritten by multiplying the exponents.
NB: W hen working with simple expressions, be sure to distribute the outside exponent to every term within the expression!
(2x3)^2 = 2^2x^6 = 4x^6
NB: When working with complex expressions, be sure to combine the terms within the parentheses before multiplying the exponents.
(2^2 +2^2)^3=(4+4)^3 =8^3 = 512
NB: If the terms within a complex expression cannot be combined, the expression must be multiplied out in its entirety.
(x^2 +y^2)^2 = (x^2 +y^2)(x^2 +y^2)
NO: (x^2 +y^2)^2 =x^4 +y^4
Exponent rule #7: Fractional exponents
To simplify an expression with a fractional exponent:
(1) First look at the bottom of the fraction. The denominator of a fractional exponent indicates what
root to take of the original base.
(2) Then look at the top of the fraction. The numerator of a fractional exponent indicates the power to which the new base should be raised.
For example, the expression 4^(3/2) = 8, since:
The denominator 2 indicates that the square root of the original base should be taken, square root of 4 = 2.
The numerator 3 indicates that the new base of 2 should be cubed, and 2^3 = 8.
To solve problems that involve exponential expressions on both sides of an equation, you must _______ (2)
1) rephrase the expressions so that both expressions either have the same base or the same exponent
2) eliminate whatever is the same – whether it’s the bases or the
exponents
(Usually breaking down the bases)
If exponents of your equation are already the same _______
you simply need to eliminate the exponents
(9z – 1)^4x = (3z + 5)^4x
9z – 1 = 3z + 5
Inequalities that have exponential expressions on both sides of their inequality signs should be solved in the same manner as equations:
Just remember to switch the direction of your inequality sign if you multiply or divide both sides of your inequality by a negative number!
Addition or Subtraction in an Exponent: Work in Reverse! (Advanced)
Difficult exponent problems often require exponent rules to be manipulated in reverse.
When given an exponent that contains addition or subtraction, it can be helpful to rewrite the term by reversing the rule!
When given an exponent that contains addition or subtraction, it can be helpful to __________
rewrite the term by reversing the rule!
3^x – 3^x-1 = 54
3^x - 3^x / 3^1 = 54
Large Bases or Exponents: Break ‘Em Down! (Advanced)
Numbers that contain large bases or large exponents, such as 10010, 962, or 5100, can be rewritten in two ways:
1) The base can be split.
Base-splitting is the “same exponent” portion of exponent rule II, used in reverse:
2) The exponent can be split
Exponent-splitting is the “same base” portion of exponent rule II, used in reverse
In most cases, it will be the bases that need to be simplified
Fractional Bases
Advanced
When given an exponential expression with a fractional base, it can be very helpful to rewrite the base
√2
≅ 1.4
√3
≅ 1.7
√4
= 2
Benchmark for values 2-10!
Values larger: + 0.2
Values smaller : - 0.3
√5
≅ 2.2
√6
≅ 2.4
√7
≅ 2.6
√8
≅ 2.8
√9
= 3
√10
≅ 3.2
NB= widespread misconception that the square root of a positive number has two solutions
the radical symbol only denotes the non-negative root of a number: an expression such as √4 only equals 2.
The negative root of a number is indicated by a negative radical: the negative root of 4 (i.e. –2) is indicated by the notation − √4.
Simplifying Square Roots
To simplify any square root, rewrite the root as a product of its factors inside the radical and simplify any pairs that lie within.
Simply replace any “pairs” that you find inside the radical with the same number outside the radical.
If a root has a coefficient, be sure to multiply that coefficient with any “pair” that you put outside the radical!
If a square root does not contain a “pair” of factors, it cannot be simplified!
Simplifying more advanced roots aka. difference “simple” and “complex” square roots
Simple: no addition or subtraction within the radical
» Break up the root
√(36×36)= √36× √36 = 6×6 = 36
√(20/100) = √20 / √100
= √(4*5) / 10
= 2√5 / 10
= √5 / 5
Complex: Addition or subtraction within the radical
» the terms within the radical must be combined
√(36 + 36) = √72
= √(662) = 6√2
Operations on roots
Can be added, subtracted, multiplied and divided
Adding / subtracting √
Terms with the same radical can be added or subtracted
Multiplying √
Combine the terms under a single radical and multiply
√3 * √2 = √(3*2) = √6
Break down terms before multiplying
√14 * √35 = √(1435)
= √(2775) = 7√10
Dividing √
Break down the radicals and cancel out like terms
√28 / √7
= √7*√4 / √7 = √4 = 2
! Whole numbers should be divided separately
8√10 / 2√5
= 8/2 * √5*√2 / √5 = 4√2
Radicals in the Den : Get rid of ‘Em
Fractions that contain radicals in their denominators are considered unsimplified.
Multiply the top and bottom of the fraction by the square root that appears in the den.
5/√7 = 5√7 / √7√7 = 5√7 / 7
» cosmetic procedure, doesn’t change value
Approximating Roots
Find the sures that it lies between and estimate.
If a root has a coefficient, it is better to convert the coefficient to a root.
Conjugates (Advanced)
√
“Complex” denominators that contain radicals cannot be simplified by the technique demonstrated in the “Radicals in the Denominator” section.
Such denominators can only be simplified by multiplying them by their conjugates, which are formed by switching their plus or minus signs.
> > Factor the difference between perfect squares into a^2 - b^2 to simplify
Roots as Exponents (Adv)
All roots can be expressed as fractional exponents.
The reverse of this process can be used to express a root as a fractional exponent. Just let the root become the denominator of the fraction.
Factoring “Complex” √ (Adv)
“Complex” square roots can be simplified by factoring out terms in common to both terms within the radical.
> > typically require the terms inside the radical to be factored. The key to solving such problems is to look for a perfect square
Roots & decimals (Adv)
In most instances, the easiest way to take the square root of a decimal is to ask yourself: “what times what would equal the square root?”
> > calculating the appropriate root, regardless of decimals
Decimals Places of Answer × “Root Number” = Decimals Places of Original
3√0.000125 = 0.05.
If the “root number” of a cube root is 3 and the number of decimal places
in the original is 6, the number of the decimal places in the answer must be 2, as:
Decimals Places of Answer × “Root Number” = Decimals Places of Root
x×3 = 6
x = 2
Therefore, if the answer has 2 decimal places, and 3√125 = 5, the answer must be 0.05.
