Download Grade 7 Exponents and Powers

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Grade 7
Exponents and Powers
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Answer t he quest ions
(1)
If x=3 and y=2, f ind the value of
(2)
(3)
Kareem plants a rose on his 10th birthday. If the plant has one rose to start with, and number of
roses triples every week, then how many roses will be there af ter x weeks?
(4) Solve f ollowing and write in simplest f raction f orm,
(5)
=?
(6) Find value of f ollowing
(7)
A)
(-2)5
B)
26
C)
(-8)2
D)
(-10)2
Find the product of square of
3
and cube of
4
(8)
1
3
What is the remainder when 26 4 is divided by 5?
(9)
=?
Choose correct answer(s) f rom given choice
(10) Simplif y the f ollowing and write the answer in exponential f orm
17 3 × 17 9 × 17 2 = ?
a. 17 14
b. 17 15
c. 17 12
d. 17 13
(11) Simplif y f ollowing and write answer in exponential f orm
259 × 257 = ?
a. 532
b. 533
c. 531
d. 534
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(12) (-4)2 × (-4)2 = ?
a. 256
b. -256
c. 260
d. 252
Fill in t he blanks
(13) Find value of f ollowing
A)
=
C)
=
B)
=
D)
=
(14) If 3x × (-2)3 = -24,
x=
Check True/False
(15) 1327 < 27 13
T rue
False
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ID : ae-7-Exponents-and-Powers [3]
Answers
(1)
512
27
Step 1
If you look at the question caref ully, you will notice that you have to f ind out the value of (y
+
y
)x at x = 3 and y = 2.
x
Step 2
Now put x = 3 and y = 2 in (y +
y
)x
x
y
(y +
)x = (2 +
x
=(
2
)3
3
6+2
)3
3
=(
8
)3
3
=
512
27
Step 3
T heref ore the value of (y +
y
x
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)x at x = 3 and y = 2 is
512
.
27
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ID : ae-7-Exponents-and-Powers [4]
(2)
1
8
Step 1
If you look at the question caref ully, you will notice that
you have to f ind out the value of (
-1
)3 × (
2
-4
)3 .
4
Step 2
By solving each term separately
(
-1
)3 =
2
-1
8
and
(
-4
)3 =
4
-64
64
Step 3
Now (
-1
2
=
)3 × (
-4
)3 =
4
-1
×
-64
8
64
1
8
Step 4
T heref ore the value of (
-1
2
(3)
)3 × (
-4
)3 is
4
1
.
8
3x
Step 1
If you look at the question caref ully, you will notice that Kareem plants a rose on his 10th
birthday. If the plant has one rose to start with, and number of roses triples every week.
Step 2
Number of rose at starting
= 1
Number of rose after 1 week = 31
Number of rose after 2 week = 32
Number of rose after 3 week = 33
----------------------------------------------------------------Number of rose after x week = 3x
Step 3
T heref ore the total number of roses af ter x weeks will be 3x.
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ID : ae-7-Exponents-and-Powers [5]
(4)
1
2
Step 1
{(
1
3
)2 + (
1
)0 }-1 can be simplif ied as:
2
1
(
1
)2 + (
1
3
)0
2
1
=
(1)2
+
(1)2
(3)0
(2)0
1
=
1
1
+
1
1
1
=
2
1
=
1
2
Step 2
Now the simplest f raction f orm of {(
1
1
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)2 + (
3
2
)0 }-1 is
1
.
2
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ID : ae-7-Exponents-and-Powers [6]
(5)
-1
2
Step 1
Rules of the exponent:
Rule 1: When you multiply two numbers with the same base, you add the exponents.
Rule 2: When you divide two numbers with the same base, you subtract the exponents.
Rule 3: When you have an exponent expression that is raised to a power, you can multiply
the exponent and power.
Rule 4: (x)-n can be written as
1
(x)n
Step 2
Now [ { (-
1
)1}1]1 can be simplif ied as:
2
1
= { (-
)1} 1 × 1
2
1
={ (-
)1}1
2
1
= (-
) 1×1
2
=(-
1
)1
2
=
-1
2
Step 3
T heref ore the value of [ { (-
1
)1}1]1 is
2
(6)
A)
-1
.
2
-32
Step 1
(-2)5 can be written as:
(-2)5 = (-2) × (-2) × (-2) × (-2) × (-2) = -32
Step 2
Now the value of (-2)5 is -32.
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B)
64
Step 1
26 can be written as:
26 = 2 × 2 × 2 × 2 × 2 × 2 = 64
Step 2
Now the value of 26 is 64.
C)
64
Step 1
(-8)2 can be written as:
(-8)2 = (-8) × (-8) = 64
Step 2
Now the value of (-8)2 is 64.
D)
100
Step 1
(-10)2 can be written as:
(-10)2 = (-10) × (-10) = 100
Step 2
Now the value of (-10)2 is 100.
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(7)
1
48
Step 1
If you look at the question caref ully, you will notice that
you have to f ind out the product of square of
3
and cube of
4
1
.
3
Step 2
square of
3
=(
4
cube of
1
3
9
)2 =
4
=(
3
1
16
1
)3 =
3
27
Step 3
Now product of
9
and
16
9
×
16
1
27
=
1
27
9
432
Step 4
T heref ore the product of square of
3
4
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and cube of
1
3
is
1
.
48
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ID : ae-7-Exponents-and-Powers [9]
(8)
1
Step 1
Let us try to observe the pattern of remainders when we divide the number by 5 on smaller
power ranges.
Number Dividend Divisor Remainder
21
2
5
2
22
4
5
4
23
8
5
3
24
16
5
1
25
32
5
2
26
64
5
4
27
128
5
3
28
256
5
1 .......
If we see closely, we notice that when we divide 2n by 5, the remainder is repeated af ter
every 4 th power of 2.
Step 2
Hence, if we divide 64 (the power of 2 in question) by 4, the remainder will tell us what
power to use to simplif y the problem.
Step 3
When we divide 64 by 4, we get the remainder of 4.
T heref ore we should use 24 to simplif y the problem.
Step 4
T his implies that when 26 4 is divided by 5, we will get the same remainder as we get when 24
is divided by 5. Which is 1.
Step 5
T hus, the remainder when 26 4 is divided by 5 is 1.
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ID : ae-7-Exponents-and-Powers [10]
(9)
-9
256
Step 1
If you look at the question caref ully, you will notice that
you have to f ind out the value of (
-1
)3 ÷ (
4
-2
)2 .
3
Step 2
By solving each term separately
(
-1
)3 =
4
-1
64
and
(
-2
)2 =
3
4
9
Step 3
Now (
-1
)3 ÷ (
4
=
-1
64
=
×
-2
)2 =
3
-1
÷
64
4
9
9
4
-9
256
Step 4
T heref ore the value of (
-1
4
)3 ÷ (
-2
)2 is
3
-9
.
256
(10) a. 17 14
Step 1
When you multiply two numbers with the same base, you add the exponents and when you
divide two numbers with the same base, you subtract the exponents.
Step 2
Now 17 3 × 17 9 × 17 2 can be simplif ied as:
17 3 × 17 9 × 17 2
= 17 (3 + 9 + 2)
= 17 14
Step 3
T heref ore the answer of 17 3 × 17 9 × 17 2 in exponential f orm is 17 14 .
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(11) a. 532
Step 1
Rules of the exponent:
Rule 1: When you multiply two numbers with the same base, you add the exponents.
Rule 2: When you divide two numbers with the same base, you subtract the exponents.
Rule 3: When you have an exponent expression that is raised to a power, you can multiply
the exponent and power.
Step 2
First of all you have to simplif y each number with the same base.
Now 259 × 257 can be simplif ied as:
259 × 257
= (5 × 5)9 × (5 × 5)7
= (52)9 × (52)7
= 518 × 514
= 5(18 + 14)
= 532
Step 3
T heref ore the value of 259 × 257 in exponential f orm is 532.
(12) a. 256
Step 1
(-4)2 × (-4)2 can be written as:
(-4)2 × (-4)2 = ((-4) × (-4)) × ((-4) × (-4))
= (16) × (16)
= 256
Step 2
Now the value of (-4)2 × (-4)2 is 256.
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ID : ae-7-Exponents-and-Powers [12]
(13) A)
-27
8
Step 1
If you look at the question caref ully, you will notice that
you have to f ind out the value of (
3
)2 × (
2
-3
)3 × (
2
2
)2 .
3
Step 2
By solving each term separately
(
3
)2 =
2
(
9
,
4
-3
)3 =
-27
2
8
and
(
2
)2 =
3
4
9
Step 3
Now (
3
2
=
)2 × (
-3
)3 × (
2
2
)2 =
3
9
4
×
-27
×
8
4
9
-27
8
Step 4
T heref ore the value of (
3
2
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)2 × (
-3
2
)3 × (
2
3
)2 is
-27
.
8
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ID : ae-7-Exponents-and-Powers [13]
B)
-1
512
Step 1
If you look at the question caref ully, you will notice that
you have to f ind out the value of (
1
)3 × (
2
-1
)3 × (
2
1
)3 .
2
Step 2
By solving each term separately
(
1
)3 =
2
(
1
,
8
-1
)3 =
-1
2
8
and
(
1
)3 =
2
1
8
Step 3
Now (
1
2
=
)3 × (
-1
)3 × (
2
1
)3 =
2
1
8
×
-1
×
8
1
8
-1
512
Step 4
T heref ore the value of (
1
2
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)3 × (
-1
2
)3 × (
1
2
)3 is
-1
.
512
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ID : ae-7-Exponents-and-Powers [14]
C)
-27
8
Step 1
If you look at the question caref ully, you will notice that
you have to f ind out the value of (
3
)2 × (
2
-3
)3 × (
2
2
)2 .
3
Step 2
By solving each term separately
(
3
)2 =
2
(
9
,
4
-3
)3 =
-27
2
8
and
(
2
)2 =
3
4
9
Step 3
Now (
3
2
=
)2 × (
-3
)3 × (
2
2
)2 =
3
9
4
×
-27
×
8
4
9
-27
8
Step 4
T heref ore the value of (
3
2
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)2 × (
-3
2
)3 × (
2
3
)2 is
-27
.
8
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ID : ae-7-Exponents-and-Powers [15]
D)
-1
18
Step 1
If you look at the question caref ully, you will notice that
you have to f ind out the value of (
1
)3 × (
2
2
)2 × (
3
-2
)3 .
2
Step 2
By solving each term separately
(
1
)3 =
2
(
1
,
8
2
)2 =
3
4
9
and
(
-2
)3 =
2
-8
8
Step 3
Now (
1
2
=
)3 × (
2
)2 × (
3
-2
)3 =
2
1
8
×
4
×
9
-8
8
-1
18
Step 4
T heref ore the value of (
1
2
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)3 × (
2
3
)2 × (
-2
2
)3 is
-1
.
18
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(14)
1
Step 1
If you look at the question caref ully, you will notice that you have to f ind out the value of x
f rom 3x × (-2)3 = -24
Step 2
Now 3x × (-2)3 = -24
⇒ 3x × -8 = -24
⇒ 3x =
-24
-8
⇒ 3x = 3
⇒ 3x = 31
⇒x=1
Step 3
T heref ore the value of x is 1.
(15) False
If you look at 1327 < 27 13 caref ully, you will notice that 1327 is greater 27 13 and hence the
statement "1327 < 27 13" is False.
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