7x Multiple (3x sq - ax)
Plz solve this problem
Answers
Rule 1. Same base
aman = am + n
"To multiply powers of the same base, add the exponents."
For example, a2a3 = a5.
Why do we add the exponents? Because of what the symbols mean. Section 1.
Example 1. Multiply 3x2 · 4x5 · 2x
Solution. The problem means (Lesson 5): Multiply the numbers, then combine the powers of x :
3x2 · 4x5 · 2x = 24x8
Two factors of x -- x2 -- times five factors of x -- x5 -- times one factor of x, produce a total of 2 + 5 + 1 = 8 factors of x : x8.
Problem 1. Multiply. Apply the rule Same Base.
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a) 5x2 · 6x4 = 30x6 b) 7x3 · 8x6 = 56x9
c) x · 5x4 = 5x5 d) 2x · 3x · 4x = 24x3
e) x3 · 3x2 · 5x = 15x6 f) x5 · 6x8y2 = 6x13y2
g) 4x · y · 5x2 · y3 = 20x3y4 h) 2xy · 9x3y5 = 18x4y6
i) a2b3a3b4 = a5b7 j) a2bc3b2ac = a3b3c4
k) xmynxpyq = xm + pyn+ q l) apbqab = ap + 1bq + 1
Problem 2. Distinguish the following:
x · x and x + x.
x · x = x². x + x = 2x.
Example 2. Compare the following:
a) x · x5 b) 2 · 25
Solution.
a) x · x5 = x6
b) 2 · 25 = 26
Part b) has the same form as part a). It is part a) with x = 2.
One factor of 2 multiplies five factors of 2 producing six factors of 2.
2 · 2 = 4 is not correct here.
Problem 3. Apply the rule Same Base.
a) xx7 = x8 b) 3 · 37 = 38 c) 2 · 24 · 25 = 210
d) 10 · 105 = 106 e) 3x · 36x6 = 37x7
Problem 4. Apply the rule Same Base.
a) xnx2 = xn + 2 b) xnx = xn + 1
c) xnxn = x2n d) xnx1 − n = x
e) x · 2xn − 1 = 2xn f) xnxm = xn + m
g) x2nx2 − n = xn + 2
Rule 2: Power of a product of factors
(ab)n = anbn
"Raise each factor to that same power."
For example, (ab)3 = a3b3.
Why may we do that? Again, according to what the symbols mean:
(ab)3 = ab · ab · ab = aaabbb = a3b3.
The order of the factors does not matter:
ab · ab · ab = aaabbb.
Problem 5. Apply the rules of exponents.
a) (xy)4 = x4y4 b) (pqr)5 = p5q5r5 c) (2abc)3 = 23a3b3c3
d) x3y2z4(xyz)5 = x3y2z4 · x5y5z5 Rule 2.
= x8y7z9 Same Base.
Rule 3: Power of a power
(am)n = amn
"To take a power of a power, multiply the exponents."
For example, (a2)3 = a2 · 3 = a6.
Why do we do that? Again, because of what the symbols mean:
(a2)3 = a2a2a2 = a3 · 2 = a6
Problem 6. Apply the rules of exponents.
a) (x2)5 = x10 b) (a4)8 = a32 c) (107)9 = 1063
Example 3. Apply the rules of exponents: (2x3y4)5
Solution. Within the parentheses there are three factors: 2, x3, and y4. According to Rule 2 we must take the fifth power of each one. But to take a power of a power, we multiply the exponents. Therefore,
(2x3y4)5 = 25x15y20
Problem 7. Apply the rules of exponents.
a) (10a3)4 = 10,000a12 b) (3x6)2 = 9x12
c) (2a2b3)5 = 32a10b15 d) (xy3z5)2 = x2y6z10
e) (5x2y4)3 = 125x6y12 f) (2a4bc8)6 = 64a24b6c48
Problem 8. Apply the rules of exponents.
a) 2x5y4(2x3y6)5 = 2x5y4 · 25x15y30 = 26x20y34
b) abc9(a2b3c4)8 = abc9 · a16b24c32 = a17b25c41
Problem 9. Use the rules of exponents to calculate the following.
a) (2 · 10)4 = 24 · 104 = 16 · 10,000 = 160,000
b) (4 · 102)3 = 43 · 106 = 64,000,000
c) (9 · 104)2 = 81 · 108 = 8,100,000,000
The powers of 10 have as many 0's as the exponent of 10.
Example 4. Square x4.
Solution. (x4)2 = x8.
To square a power, double the exponent.
Problem 10. Square the following.
a) x5 = x10 b) 8a3b6 = 64a6b12
c) −6x7 = 36x14 d) xn = x2n
Part c) illstrates: The square of a number is never negative.
(−6)(−6) = +36. The Rule of Signs.
Problem 11. Apply a rule of exponents -- if possible.
a) x2x5 = x7, Rule 1. b) (x2)5 = x10, Rule 3.
c) x2 + x5
Not possible. The rules of exponents apply only to multiplication.
In summary: Add the exponents when the same base appears twice: x2x4 = x6. Multiply the exponents when the base appears once -- and in parentheses: (x2)5 = x10.
Problem 12. Apply the rules of exponents.
a) (xn)n = xn · n = xn2 b) (xn)2 = x2n
Problem 13. Apply a rule of exponents or add like terms -- if possible.
a) 2x2 + 3x4 Not possible. These are not like terms.
b) 2x2 · 3x4 = 6x6. Rule 1.
c) 2x3 + 3x3 = 5x3. Like terms. The exponent does not change.
d) x2 + y2 Not possible. These are not like terms.
e) x2 + x2 = 2x2. Like terms.
f) x2 · x2 = x4. Rule 1
g) x2 · y3 Not possible. Different bases.
h) 2 · 26 = 27. Rule 1
i) 35 + 35 + 35 = 3 · 35 (On adding those like terms) = 36.