9t2 - 3/2+ 1/16 when t is 2
Answers
Answer:
Step by Step Solution:
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STEP
1
:
1
Simplify ——
16
Equation at the end of step
1
:
3 1
((9•(t2))+(—•t))+——
2 16
STEP
2
:
3
Simplify —
2
Equation at the end of step
2
:
3 1
((9 • (t2)) + (— • t)) + ——
2 16
STEP
3
:
Equation at the end of step
3
:
3t 1
(32t2 + ——) + ——
2 16
STEP
4
:
Rewriting the whole as an Equivalent Fraction
4.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 2 as the denominator :
32t2 32t2 • 2
32t2 = ———— = ————————
1 2
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
32t2 • 2 + 3t 18t2 + 3t
————————————— = —————————
2 2
Equation at the end of step
4
:
(18t2 + 3t) 1
——————————— + ——
2 16
STEP
5
:
STEP
6
:
Pulling out like terms
6.1 Pull out like factors :
18t2 + 3t = 3t • (6t + 1)
Calculating the Least Common Multiple :
6.2 Find the Least Common Multiple
The left denominator is : 2
The right denominator is : 16
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
2 1 4 4
Product of all
Prime Factors 2 16 16
Least Common Multiple:
16
Calculating Multipliers :
6.3 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 8
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
6.4 Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. 3t • (6t+1) • 8
—————————————————— = ———————————————
L.C.M 16
R. Mult. • R. Num. 1
—————————————————— = ——
L.C.M 16
Adding fractions that have a common denominator :
6.5 Adding up the two equivalent fractions
3t • (6t+1) • 8 + 1 144t2 + 24t + 1
——————————————————— = ———————————————
16 16
Trying to factor by splitting the middle term
6.6 Factoring 144t2 + 24t + 1
The first term is, 144t2 its coefficient is 144 .
The middle term is, +24t its coefficient is 24 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 144 • 1 = 144
Step-2 : Find two factors of 144 whose sum equals the coefficient of the middle term, which is 24 .
-144 + -1 = -145
-72 + -2 = -74
-48 + -3 = -51
-36 + -4 = -40
-24 + -6 = -30
-18 + -8 = -26
-16 + -9 = -25
-12 + -12 = -24
-9 + -16 = -25
-8 + -18 = -26
-6 + -24 = -30
-4 + -36 = -40
-3 + -48 = -51
-2 + -72 = -74
-1 + -144 = -145
1 + 144 = 145
2 + 72 = 74
3 + 48 = 51
4 + 36 = 40
6 + 24 = 30
8 + 18 = 26
9 + 16 = 25
12 + 12 = 24 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 12 and 12
144t2 + 12t + 12t + 1
Step-4 : Add up the first 2 terms, pulling out like factors :
12t • (12t+1)
Add up the last 2 terms, pulling out common factors :
1 • (12t+1)
Step-5 : Add up the four terms of step 4 :
(12t+1) • (12t+1)
Which is the desired factorization
Multiplying Exponential Expressions:
6.7 Multiply (12t+1) by (12t+1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (12t+1) and the exponents are :
1 , as (12t+1) is the same number as (12t+1)1
and 1 , as (12t+1) is the same number as (12t+1)1
The product is therefore, (12t+1)(1+1) = (12t+1)2
Final result :
(12t + 1)2
——————————
16
Answer:
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