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```Scientific notation
1. Write the following numbers in standard decimal form.
a) 1 × 103 =
b) 2.3 × 102 =
c) 6.44 × 105 =
d) 1.446 × 102 =
e) 3.8 × 10−2 =
f) 4.29 × 10−4 =
g) 1.63 × 10−1 =
h) 2.7 × 100 =
2. Write the following numbers in scientific notation.
a) 12
b) 247
c) 20 000
d) 12 121
e) 61 000 000
f)
5
g) 0.000 000 1
h) 0.335
i)
0.000 129
j) 0.6
(Hint: Change to standard decimal form, do the calculation and then change back to
scientific notation.)
a) 5.33 × 102 + 3.771 × 103
b) 1 × 103 + 2 × 102
c) 7.44 × 105 − 9.67 × 104
d) 2.47 × 10−3 + 1.6 × 10−2
e) 1.6 × 10−2 + 6.5 × 10−3
Index laws: Product rule
1. Identifying index notation
Read the following expressions carefully. Which are not written using index notation?
a) 103
d) 8 x 8 x 8
b) 2 000
e) 1210
c) 32
f)
10 x 24
2. Simplifying multiplications to index notation
Example: 2 x 2 x 2 = 23
Simplify the following expressions using index notation.
a) 8 x 8
g) –9 x –9
b) 3 x 3 x 3 x 3
c) 11 x 11 x 11 x 11 x 11 x 11 x 11 x
11 x 11
h) 33 x 33 x 33
i) –6 x –6 x –6 x –6
d) 7 x 7 x 7
e) 100 x 100 x 100 x 100 x 100
k) 30 x 30 x 30 x 30
f)
j)
l)
9x9x9x9x9x9x9x9x9
14 x 14
23 x 23 x 23 x 23 x 23 x 23 x 23 x
23 x 23 x 23 x 23 x 23
3. Write the following numbers in expanded notation:
a) 43
f)
98
k) (–123)2
b) 110
g) 56
l)
c) 122
h) 118
m) (20)2
d) 554
i)
33
n) (12)3
e) 62
j)
(–2)4
o) (305)2
(–6)3
4. Simplify the following expressions using index notation:
a) 15 x 15 x 4 x 4 x 15
h) 8 x –2 x 8 x 8 x –2 x 12 x 12 x 12 x 12
b) 3 x 4 x 3 x 4
i)
12.4 x 3 x 12.4 x 12.4 x –17 x –17 x 3
c) 9 x 9 x 9 x 9 x 6 x 9 x 6
j)
9x4x4
d) 3 x 2 x 2 x 2 x 2 x 2 x 3 x 3
k) 3 x 3 x –8 x –16 x –8 x 3
e) –4 x 7 x –4 x 7 x 7 x 7 x 7
l)
f)
–2 x 3 x 3 x –2
g) 10 x –6 x 10 x –6 x 8 x 8 x 10
10 x 10 x 14 x 15 x 14
m) 2 x 9 x 14 x 3 x 14 x 8
n) 124 x 124 x 2 x 124 x 4
1. Simplify the following expressions using the first index law.
(ab)8 x (ab)3
k) (–d)2 x (–d)4
a) 43 x 42
f)
b) 2 x 24
g) 66 x 63 x 62
l)
c) 3.24 x 3.25
h) b8 x b2 x b
m) 12 x 12 x 12
d) 554 x 5510
i)
0.33 x 0.35
n) (–3)3 x (–3)1
e) b2 x b5
j)
(–2)4 x (–2)3
o) r2 x r2 x r2 x r2
y3 x y1 x y3
2. Multiplying with exponents
a) 10f x 10f
d) 4z3 x 6z5
g) –10e8 x –3e5
b) 3h x 4h
e) 15d3 x 4d2
h) t x –t4
c) 9m x 2m2
f)
–4r2 x 2r4
3p x 4p8 x p2
i)
3. Dividing powers with the same base
Rewrite the following as a single power, using the second index law
54
a) 52
b)
910
96
330
c) 313
147
d) 145
e)
2011
204
0.54
f) 0.52
4. Dividing powers with the same pronumerals as the base
Rewrite the following as a single power, using the second index law.
𝑎5
a) 𝑎2
b)
𝑏20
𝑏6
𝑐4
c) 𝑐 2
d)
𝑑 14
𝑑8
e)
f)
𝑒 100
𝑒 80
𝑓4
𝑓
Index laws: Power of a product or quotient rules
1. Use the general rule (ab)m = ambm to simplify the following powers of products.
a) (8 × 9)3
d) (𝑒 × 5)6
b) (11 × 3)4
e) (6𝑓)8
c) (−1 × 7)6
f) (−2𝑔)6
2. Expand the following powers or quotients.
3 3
a) (5)
10 2
b) ( 2 )
−2 4
c) ( 3 )
𝑎 2
d) (5 )
3 4
e) (𝑏)
−𝑏 3
f) ( 𝑐 )
3. Select the appropriate strategy and simplify the following powers:
𝑎 2
4
a) ( )
−1 3
b) ( 7 )
c) (18 × 2)4
d) (𝑎 × 3)5
e) (6𝑏)10
5 5
f) (11)
g) (5𝑑)4
14 8
𝑓
h) ( )
23
3
i) ( 𝑏 )
𝑑 2
4
j) ( 5 )
k) (𝑎𝑏)4 × (𝑎𝑏)5
6 5
6 4
l) (2) × (2)
Index laws: Zero and power of a power rules
1. To the power of zero:
Calculate the value of each of the following powers.
a) 100
e) b0
b) 2230
f) (gh)0
c) –30
g) 142 529 723 478 234.2340
d) 0.240
h) (abcdefg)0
2. Simplify the following expressions using the power of a power index law.
a) (53)2
d) (3.23)5
b) (24)5
e) (552)2
c) (–103)3
f) (0.74)2
3. Simplify these expressions:
a) (b2)5
e) ((3c)3)2
b) (g4)3
f) (43)a
c) (ha)2
g) (ab)c
d) ((ab)8)3
h) ((de)f)g
4. Simplify these expressions. Use any of the index laws you know.
a) (66 x 63)2
128
b) (122)3
c)
(a5 x
a-2) 3
d) (b-5 x b4)-2
e) (10-3 x 103)11
𝑔3
f) (𝑔2)6
Expanding binomials questions
1. Expand the following
a) 5(𝑎 + 3)
b) 2(5 + 𝑏)
c) 𝑐(2 + 10)
d) 4(𝑑 − 3)
e) 𝑒(6 − 4)
f) 3(𝑓 + 2)
a) 5𝑎(𝑎 + 2)
b) 3𝑥(2 + 𝑏)
c) 𝑐 2 (2𝑥 + 3)
d) 4𝑦(4 + 𝑑)
e) −𝑒(2 − 8𝑒)
f) −3𝑥(−𝑓 + 2)
a) (3 + 𝑥)(𝑎 + 2)
b) (4 + 𝑏)(2 + 𝑏)
c) (2𝑥 + 3)(𝑐 − 2)
d) (5𝑑 + 2)(4 + 𝑑)
e) (3 − 𝑒)(2 − 8𝑒)
f) (−𝑓 + 2)(𝑓 + 3)
2. Expand the following
3. Expand the following binomials
4. Expand the following perfect squares
a) (𝑎 + 2)(𝑎 + 2)
b) (4 + 𝑏)(4 + 𝑏)
c) (2𝑥 + 3)(2𝑥 + 3)
d) (4 − 𝑑)2
e) (2 − 3𝑒)2
f) (−𝑓 + 2)2
Calculating simple interest
a) Kaylee needed \$500 for
concert tickets, so she
borrowed that amount from a
bank. She had to repay the
money in two years and was
charged 10% interest.
b) Marina got a loan for
10 years at 6.5%. She had
some savings, so only needed
an extra \$12 000 to buy the
car she wanted.
c) At 7.6% for 20 years, Mal
thought he had a great loan to
get the \$15 000 he needed to
start a company.
d) Simon was studying
medicine at university
and he needed \$800 to
school supplies. He
found a loan which
charged 20% and he
e) Jane borrowed \$12 000 at
6.4%. She planned on using
the money to take flying
lessons, then work as a pilot
to get the money to pay it back
in eight years.
f) Dan needed to borrow
money for a year before an
would be paid. He borrowed
\$800 to help cover the costs
of his classes until then. The
bank offered him a 9% loan.
Questions 1 to 3 are about the situations above.
1. Identify the principal, rate and time for each of the situations above.
2. Calculate the simple interest owed.
3. Calculate the total repayment needed to repay the loan. (Total repayment = principal +
simple interest.)
4. Complete the table below by solving the unknown variable in each row.
Simple interest
Principal
Rate
Time
\$300
8%
3 years
0.05
4 years
\$125
\$192
\$800
\$5 400
\$12 000
5 years
\$800
\$2 000
20 years
\$150.75
450
\$100
12%
6.7%
0.10
10 years
Factorize each linear expression.
1)
6x + 9
2)
–20y – 5z
3)
15 – 3a
4)
2m + 2
5)
39u – 52v + 13
6)
10z – 60
7)
44p + 11q
8)
42 + 35w
9)
81n – 36
10)
40b – 80c – 40d
Solving worded questions using linear equations
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