What are the rules for simplifying exponents? • Whenever you multiply two terms with the same base, you can add the exponents: EX: ( x m ) ( x n ) = x( m + n ) (((However, we can NOT simplify (x4)(y3), because the bases are different: (x4)(y3) = xxxxyyy = (x4)(y3). Nothing combines.)) • Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power: ( xm ) n = x m n • If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance, (xy2)3 = (xy2)(xy2)(xy2) = (xxx)(y2y2y2) = (xxx)(yyyyyy) = x3y6 = (x)3(y2)3. Another example would be:
((((This rule does NOT work if you have a sum or difference within the parentheses. Exponents, unlike mulitiplication, do NOT "distribute" over addition. For instance, given (3 + 4)2, do NOT say "This equals 32 + 42 = 9 + 16 = 25", because this is wrong. Actually, (3 + 4)2 = (7)2 = 49, not 25.))) • Anything to the power zero is just "1". A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x–2" (ecks to the minus two) just means "x2, but underneath, as in 1/(x2)".
Further Explanation and examples at the following website < http://www.purplemath.com/modules/simpexpo.htm>
What are the rules for simplifying exponents? -There are a good bit of rules for simplifying exponents, and all of them are fairly easy. You need to know the rules for simplifying exponents for what we learned yesterday in chapter five. In order to do the problem you would have to know the exponent rules in order to make the problem simpler to work.
Exponent Rules:
Product Rule: Simplify (x^3)(x^4) -When multiplying, add he exponents- (x^m)(x^n)=x(m+n)
Power Rule: Simplify (x^2)^4 -The "power rule" tells us that to raise a power to a power, just multiply the exponents.
*Anything to the power zero is just 1.
Quotient Rule: x^m/x^n=x^m-n -The quotient rule tells us that we can divide two powers with the same base by subtracting the exponent.
Negative Exponents: 4^-2=1/4^2=1/16 -Tells us that any nonzero number raised to a negative power equals is reciprocal raised to the opposite positive power.
What are the rules for simplifying logs? -This also goes along with what we started learning in chapter five. In order to make the problems are simplest as possible to work, you also need to know the log rules along with the exponent rules.
LOG RULES:
-Multiplication inside the log can be turned into addition outside the log, and vice versa. *logb(mn) = logb(m) + logb(n)
-Division inside the log can be turned into subtraction outside the log, and vice versa. *logb(m/n) = logb(m) – logb(n)
-An exponent on everything inside a log can be moved out front as a multiplier, and vice versa *logb(mn) = n · logb(m)
-whenever you multiply two terms with the SAME BASE (x^m)(x^n) = x^(m+n) EX: (B^3)(B^2)= B^5
-whenever you divide two terms with the SAME BASE (x^m)/(x^n) = x^(m-n) EX: (B^4)/(B^2)= B^2
-when TWO bases are being multiplied and raised to the same exponent ((a)(b))^x = (a^x)(b^x) EX: (XY)^3 = (X^3)(Y^3)
-when TWO bases are being divided and raised to the same exponent (a/b)^x = (a^x)/(b^x) EX: (X/Y)^2 = (X^2)/(Y^2)
-when a BASE AND ITS EXPONENT are raised to an exponent (b^x)^y = b^(x)(y) EX: (X^3)^2 = X^6
-when a base is raised to a FRACTIONAL EXPONENT b^(x/y) = y-root (b^x) EX: A^(X/Y) = Y-root (A^X)
-TO SOLVE FOR AN EXPONENT: a. write as the same base b. set exponents equal c. solve for x EX: 2^X= (1/8)=> 2^X=8^(-1) 2^X = 2^(-3) X = -3
What are the rules for simplifying logs?
-when expanding or condensing logs, multiplication and addition go hand-in-hand. log MN= log M + log N EX: log 3N = log 3 + log N
-when expanding or condensing logs, division and subtraction go together log M/N = log M - log N EX: log 3/N = log 3 - log N
-when logs are raised to an exponent, the exponent goes to the front log M^N = N log M EX: log 3^X = X log 3
-when a when the base of the log and the number within the log are the same, and the number within is raised to an exponent, the answer is the exponent log(sub)M(sub)M^N = N EX: log(sub)3(sub)3^N = N
-when a number is raised to a logarithmic exponent M^(log N) = N EX: 3^ (log 6) = 6
1) When multiplying exponential terms, add the exponents. (b^x)(b^y) = b^x+y
2) When dividing exponential terms, subtract the exponents. b^x / b^y = b^x-y
3) When an exponent is on the outside of parenthesizes, distribute the exponent to every term. (ab)^x = (a^x)(b^x)
4) When an entire fraction is raised to an exponent, distribute the exponent to both the top and bottom. (a/b)^x = a^x / b^x
5) When one exponent is on the outside of the parenthesizes and another is on the inside, multiply the two. (b^x)^y = b^xy
6) When a number or variable is raised to a fractional exponent, the denominator of the exponent is the root and the numerator is the exponent of the term inside the root. b^x/y = y square root of b^x
7) To solve for an exponent: a. write as the same base b. set exponents equal c. solve for x
Rules for simplifying logarithms: *the base of log is always 10 unless indicated otherwise **the base of ln (natural log) is always e *** (b) represents the base b of the log
1) If logs are added, then multiply the terms. log a + log b = log ab
2) If logs are subtracted, then divide the terms. log a – lob b = log a/b
3) When the term in a log is raised to an exponent, you can bring the exponent to the front of the log. log a^b = b log a
4) If the base and the term are the same, then x is the exponent of the term. log(b) b^a = a
5) If a term is raised to a log and is the same as the base of the log, then x is the term following the log. b^log(b) a = a
**Rules for simplifying exponents: -When multiplying exponentially expressed numbers of the same base, you add their exponents. ex: 3a^-2 x 3a^5 =3a^3 -When dividing exponentially expressed numbers of the same base, you subtract their exponents. *always the smaller number from the bigger number, no matter if it's the top or bottom. Or I suppose it doesn't really matter as long as you watch out for negatives and such. ex: 2x^4/2x^6 =1/2x^2 -When a term(s) is inside of parenthesis and raised to an exponent, you distribute the exponent to every single term. ex: (4x3)^2 =16x^6 *you do NOT add the exponents here; you multiply them -the same goes for if you have a fraction in parenthesis raised to an exponent..you raise each number or variable within the parenthesis to the exponent on the outside -again, if you have an exponent on the inside of a parenthesis and the outside, multiply the two together. ex: (5^2)^3 =5^6 -When a number/term is raised to a fractional exponent, it turns into a square root..as in, the denominator of the fraction becomes the root(square or cube root, etc) and the numerator goes inside the squareroot and the term is raised to it ex: (5)^3/2 =square root of 5^3
**Rules for simplifying logs -When the log is in the form log(b)x=a---(b) is base b by the way---in order to solve it, you must put it in exponential form, which means you basically flip the order, so the exponential form would be b^a=x ex: log(2)16=x 2^x=16 (to solve, you have to put 16 in base 2) so that would be 2^4, so you have: 2^x=2^4 now that you have the same base, you set the exponents equal and your answer is x=4 --Then, when you're given a log in exponential form that you can't simplify further, you write it in terms of a log...meaning you write log in front of it and simplify *Properties of logs: -With logs, multiplying and adding is the same thing: log(b)MN = log(b)M + log(b)N ex: log(2)xy^2 =log(2)x + 2log(2)y -Also with logs, subtracting is the same as dividing...log(b)M/N = log(b)M - log(b)N ex: log45 - 2log3 =log 45/3^2 =log 45/9 =log 5 -the expoent always is brought to the front log(b)M^k = klog(b)M *shown in previous examples -log(b)b^k...the base and term cancel leaving you with the exponent as your answer -b^log(b)k = k the b^log(b) cancels and for your answer you're left with the term
what are the rules for simplifying exponents? when multiplying exponents that have the same base you can simply add the exponents together.
for example: (x^2)(x^3) = x^5
and when dividing exponents with the same base, you subtract the exponents.
example: x^5 / x^3 = x^2
when multiplying exponents in parenthesis with a separate exponent, you do not add the terms, you multiply them. when a number is raised to a fraction you raise that number to the numerator and find the root raised to the denominator.
what are the rules for simplifying logs? when given an equation in which you must find x when it is an exponent of e, you find the natural log of each side of the equation. this cancels out e and the you simply solve for the other side. You may use the logarithm function if instead of e, x is an exponent of ten.
adding logs is the same as multiplying what is inside of them.
for example: logx + 3logx = 4logx
the same applies when subtracting logs only you divide what is inside of the log when subtracting.
rules for simplifying exponents? well, first off, I'd say, or rather ask, you to define "simplify exponents" exponents are numbers, unless you're talking about variable exponents but the rules are simple anything raised to the 0th power = 1 anything raised to the 1st power = the original number anything raised to the 2nd power = the original number times itself etc.
rules for simplifying logs? it's all about placement of variables or whatever you're doing I have no idea how to explain the rules actually, but I've got them down in my head log1 will always = 0 uhh......I still can't think of how to word the rules in an understandable way
This comment has been removed by the author.
ReplyDeleteWhat are the rules for simplifying exponents?
ReplyDelete• Whenever you multiply two terms with the same base, you can add the exponents:
EX: ( x m ) ( x n ) = x( m + n )
(((However, we can NOT simplify (x4)(y3), because the bases are different: (x4)(y3) = xxxxyyy = (x4)(y3). Nothing combines.))
• Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power:
( xm ) n = x m n
• If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance, (xy2)3 = (xy2)(xy2)(xy2) = (xxx)(y2y2y2) = (xxx)(yyyyyy) = x3y6 = (x)3(y2)3. Another example would be:
((((This rule does NOT work if you have a sum or difference within the parentheses. Exponents, unlike mulitiplication, do NOT "distribute" over addition.
For instance, given (3 + 4)2, do NOT say "This equals 32 + 42 = 9 + 16 = 25", because this is wrong. Actually, (3 + 4)2 = (7)2 = 49, not 25.)))
• Anything to the power zero is just "1".
A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x–2" (ecks to the minus two) just means "x2, but underneath, as in 1/(x2)".
Further Explanation and examples at the following website
< http://www.purplemath.com/modules/simpexpo.htm>
What are the rules for simplifying logs?
• logb(mn) = logb(m) + logb(n)
Multiplication inside the log can be turned into addition outside the log, and vice versa.
• logb(m/n) = logb(m) – logb(n)
Division inside the log can be turned into subtraction outside the log, and vice versa.
• logb(mn) = n • logb(m)
An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.
((( Just as when you're dealing with exponents, the above rules work only if the bases are the same. For instance, the expression "logd(m) + logb(n)" cannot be simplified, because the bases (the "d" and the "b") are not the same, just as x2 × y3 cannot be simplified (because the bases x and y are not the same).)))
• Simplify log2(x) + log2(y).
Since these logs have the same base, the addition outside can be turned into multiplication inside:
log2(x) + log2(y) = log2(xy)
The answer is log2(xy).
• Simplify log3(4) – log3(5).
Since these logs have the same base, the subtraction outside can be turned into division inside:
log3(4) – log3(5) = log3(4/5)
The answer is log3(4/5).
• Simplify 2log3(x). Copyright © Elizabeth Stapel 2010 All Rights Reserved
The multiplier out front can be taken inside as an exponent:
2log3(x) = log3(x2)
• Simplify 3log2(x) – 4log2(x + 3) + log2(y).
I will get rid of the multipliers by moving them inside as powers:
3log2(x) – 4log2(x + 3) + log2(y)
= log2(x3) – log2((x + 3)4) + log2(y)
Then I'll put the added terms together, and convert the addition to multiplication:
log2(x3) – log2((x + 3)4) + log2(y)
= log2(x3) + log2(y) – log2((x + 3)4)
= log2(x3y) – log2((x + 3)4)
Then I'll account for the subtracted term by combining it inside with division:
More information on this can be found at the following website
http://www.purplemath.com/modules/logrules.htm
What are the rules for simplifying exponents?
ReplyDelete-There are a good bit of rules for simplifying exponents, and all of them are fairly easy. You need to know the rules for simplifying exponents for what we learned yesterday in chapter five. In order to do the problem you would have to know the exponent rules in order to make the problem simpler to work.
Exponent Rules:
Product Rule:
Simplify (x^3)(x^4)
-When multiplying, add he exponents- (x^m)(x^n)=x(m+n)
Power Rule:
Simplify (x^2)^4
-The "power rule" tells us that to raise a power to a power, just multiply the exponents.
*Anything to the power zero is just 1.
Quotient Rule:
x^m/x^n=x^m-n
-The quotient rule tells us that we can divide two powers with the same base by subtracting the exponent.
Negative Exponents:
4^-2=1/4^2=1/16
-Tells us that any nonzero number raised to a negative power equals is reciprocal raised to the opposite positive power.
What are the rules for simplifying logs?
-This also goes along with what we started learning in chapter five. In order to make the problems are simplest as possible to work, you also need to know the log rules along with the exponent rules.
LOG RULES:
-Multiplication inside the log can be turned into addition outside the log, and vice versa.
*logb(mn) = logb(m) + logb(n)
-Division inside the log can be turned into subtraction outside the log, and vice versa.
*logb(m/n) = logb(m) – logb(n)
-An exponent on everything inside a log can be moved out front as a multiplier, and vice versa
*logb(mn) = n · logb(m)
For the Log rules:
ReplyDelete< http://www.purplemath.com/modules/simpexpo.htm>
What are the rules for simplifying exponents?
ReplyDelete-whenever you multiply two terms with the SAME BASE
(x^m)(x^n) = x^(m+n)
EX: (B^3)(B^2)= B^5
-whenever you divide two terms with the SAME BASE
(x^m)/(x^n) = x^(m-n)
EX: (B^4)/(B^2)= B^2
-when TWO bases are being multiplied and raised to the same exponent
((a)(b))^x = (a^x)(b^x)
EX: (XY)^3 = (X^3)(Y^3)
-when TWO bases are being divided and raised to the same exponent
(a/b)^x = (a^x)/(b^x)
EX: (X/Y)^2 = (X^2)/(Y^2)
-when a BASE AND ITS EXPONENT are raised to an exponent
(b^x)^y = b^(x)(y)
EX: (X^3)^2 = X^6
-when a base is raised to a FRACTIONAL EXPONENT
b^(x/y) = y-root (b^x)
EX: A^(X/Y) = Y-root (A^X)
-TO SOLVE FOR AN EXPONENT:
a. write as the same base
b. set exponents equal
c. solve for x
EX: 2^X= (1/8)=> 2^X=8^(-1) 2^X = 2^(-3)
X = -3
What are the rules for simplifying logs?
-when expanding or condensing logs, multiplication and addition go hand-in-hand.
log MN= log M + log N
EX: log 3N = log 3 + log N
-when expanding or condensing logs, division and subtraction go together
log M/N = log M - log N
EX: log 3/N = log 3 - log N
-when logs are raised to an exponent, the exponent goes to the front
log M^N = N log M
EX: log 3^X = X log 3
-when a when the base of the log and the number within the log are the same, and the number within is raised to an exponent, the answer is the exponent
log(sub)M(sub)M^N = N
EX: log(sub)3(sub)3^N = N
-when a number is raised to a logarithmic exponent
M^(log N) = N
EX: 3^ (log 6) = 6
What are the rules for simplifying exponents?
ReplyDelete(x^m)(x^n) = x^(m + n)
(xm)^n = x^(mn)
x^m / x^n = x^(m-n)
What are the rules for simplifying logs?
logbx + logby = logb(xy)
logbx - logby = logb(x/y)
n logbx = logb(x^n)
logax / logab = logbx
log a a = 1
log a 1 = 0
log a (1/x) = -log a x
log a x = log x / log a = ln x / ln a
Rules for simplifying exponents:
ReplyDelete1) When multiplying exponential terms, add the exponents.
(b^x)(b^y) = b^x+y
2) When dividing exponential terms, subtract the exponents.
b^x / b^y = b^x-y
3) When an exponent is on the outside of parenthesizes, distribute the exponent to every term.
(ab)^x = (a^x)(b^x)
4) When an entire fraction is raised to an exponent, distribute the exponent to both the top and bottom.
(a/b)^x = a^x / b^x
5) When one exponent is on the outside of the parenthesizes and another is on the inside, multiply the two.
(b^x)^y = b^xy
6) When a number or variable is raised to a fractional exponent, the denominator of the exponent is the root and the numerator is the exponent of the term inside the root.
b^x/y = y square root of b^x
7) To solve for an exponent:
a. write as the same base
b. set exponents equal
c. solve for x
Rules for simplifying logarithms:
*the base of log is always 10 unless indicated otherwise
**the base of ln (natural log) is always e
*** (b) represents the base b of the log
1) If logs are added, then multiply the terms.
log a + log b = log ab
2) If logs are subtracted, then divide the terms.
log a – lob b = log a/b
3) When the term in a log is raised to an exponent, you can bring the exponent to the front of the log.
log a^b = b log a
4) If the base and the term are the same, then x is the exponent of the term.
log(b) b^a = a
5) If a term is raised to a log and is the same as the base of the log, then x is the term following the log.
b^log(b) a = a
Simplifying exponents:
ReplyDeleteMultiplication: add the exponents
y^2xy^5=y^7
Division: subtract the exponents
y^5/y^2=y^3
Exponent on outside: distribute the exponent
(x^2)^8=x^16
Fractions: top is the root, bottom is exponent
x^4/3= 4th root of x^3
Logs: this should be a review
Logs have a natural base of 10
Natural logs have a natural base e
Addition: multiply terms
Subtracion: divide terms
Exponents: place in front of the log
EXPONENT RULES:
ReplyDeletewhen multiplying, add your exponents
x^7 * x^5 = x^12
when dividing, subtract
x^7 / x^5 = x^2
when an exponent is on the outside of another, distribute it
(x^7)^5 = x^35
when an exponent is a fraction, the top is the root, and the bottom is exponent
x^7/5 = 7th root of x^5
LOGS
logs' natural base is 10; natural logs' base is e
when:
adding-multply the terms
subtracting-divide the terms
exponents-distribute
**Rules for simplifying exponents:
ReplyDelete-When multiplying exponentially expressed numbers of the same base, you add their exponents.
ex: 3a^-2 x 3a^5
=3a^3
-When dividing exponentially expressed numbers of the same base, you subtract their exponents. *always the smaller number from the bigger number, no matter if it's the top or bottom. Or I suppose it doesn't really matter as long as you watch out for negatives and such.
ex: 2x^4/2x^6
=1/2x^2
-When a term(s) is inside of parenthesis and raised to an exponent, you distribute the exponent to every single term.
ex: (4x3)^2
=16x^6 *you do NOT add the exponents here; you multiply them
-the same goes for if you have a fraction in parenthesis raised to an exponent..you raise each number or variable within the parenthesis to the exponent on the outside
-again, if you have an exponent on the inside of a parenthesis and the outside, multiply the two together.
ex: (5^2)^3
=5^6
-When a number/term is raised to a fractional exponent, it turns into a square root..as in, the denominator of the fraction becomes the root(square or cube root, etc) and the numerator goes inside the squareroot and the term is raised to it
ex: (5)^3/2
=square root of 5^3
**Rules for simplifying logs
-When the log is in the form log(b)x=a---(b) is base b by the way---in order to solve it, you must put it in exponential form, which means you basically flip the order, so the exponential form would be b^a=x
ex: log(2)16=x
2^x=16
(to solve, you have to put 16 in base 2)
so that would be 2^4, so you have:
2^x=2^4
now that you have the same base, you set the exponents equal and your answer is x=4
--Then, when you're given a log in exponential form that you can't simplify further, you write it in terms of a log...meaning you write log in front of it and simplify
*Properties of logs:
-With logs, multiplying and adding is the same thing:
log(b)MN = log(b)M + log(b)N
ex: log(2)xy^2
=log(2)x + 2log(2)y
-Also with logs, subtracting is the same as dividing...log(b)M/N = log(b)M - log(b)N
ex: log45 - 2log3
=log 45/3^2
=log 45/9
=log 5
-the expoent always is brought to the front
log(b)M^k = klog(b)M
*shown in previous examples
-log(b)b^k...the base and term cancel leaving you with the exponent as your answer
-b^log(b)k = k
the b^log(b) cancels and for your answer you're left with the term
what are the rules for simplifying exponents? when multiplying exponents that have the same base you can simply add the exponents together.
ReplyDeletefor example: (x^2)(x^3) = x^5
and when dividing exponents with the same base, you subtract the exponents.
example: x^5 / x^3 = x^2
when multiplying exponents in parenthesis with a separate exponent, you do not add the terms, you multiply them. when a number is raised to a fraction you raise that number to the numerator and find the root raised to the denominator.
what are the rules for simplifying logs?
when given an equation in which you must find x when it is an exponent of e, you find the natural log of each side of the equation. this cancels out e and the you simply solve for the other side. You may use the logarithm function if instead of e, x is an exponent of ten.
adding logs is the same as multiplying what is inside of them.
for example: logx + 3logx = 4logx
the same applies when subtracting logs only you divide what is inside of the log when subtracting.
Use the following examples to simplify exponents
ReplyDelete-(x^a)(x^b) = x^(a + b)
-(xa)^b = x^(ab)
-x^a / x^b = x^(a-b)
Use the following to simplify logs
logcx + logcy = logc(xy)
-logcx - logcy = logc(x/y)
-b logcx = logc(x^b)
-logcx / logcb = logbx
-log c c = 1
-log c 1 = 0
-log c (1/x) = -log c x
-log c x = log x / log c = ln x / ln c
rules for simplifying exponents?
ReplyDeletewell, first off, I'd say, or rather ask, you to define "simplify exponents"
exponents are numbers, unless you're talking about variable exponents
but the rules are simple
anything raised to the 0th power = 1
anything raised to the 1st power = the original number
anything raised to the 2nd power = the original number times itself
etc.
rules for simplifying logs?
it's all about placement of variables or whatever you're doing
I have no idea how to explain the rules actually, but I've got them down in my head
log1 will always = 0
uhh......I still can't think of how to word the rules in an understandable way