Chapter 1: Algebraic Expressions

Question 1

a^m x a^n=

Answer 1

a^m+n

Question 1

a^m / a^n=

Answer 1

a^m-n

Question 2

(a^m)^n=

Answer 2

a^mn

Question 3

(ab)^n

Answer 3

a^n b^n

Question 4

For the quadratic expression, 2x^2+5x-12, describe how I can factorise.

Answer 4

Split the middle term, into 2 numbers that multiply to create constant term, and add for middle term:
e.g. 8+-3

Therefore,
2x^2+8x
-3x-12

Factorise:
2x(x+4) -3(x+4)
First bracket is common multiplier (e.g. (x+4)

Second bracket is multiples outside (e.g (2x-3))

Thus:
(x+4)(2x-3)

Question 5

a^0=

Answer 5

1

Question 6

a^1/m=

Answer 6

m√a

Question 7

a^n/m=

Answer 7

(m√a)^n

Question 8

a^-m=

Answer 8

1/a^m

Question 9

a^-n/m=

Answer 9

1/(a^m)^n

Question 10

If b=1/9a^2, determine
3b^-2 in the form ka^n, with k and n constants.

Answer 10

b=1/9a^2
Therefore 3b^-2:
3(1/9a^2)^-2
3(a^2/9)^-2
3(9/a^2)^2
3(81/a^4)
=243/a^4
=243a^-4

Question 11

If 125√5=5%k
What is k?

Answer 11

5^3 x 5^1/2
=5^7/2
k=7/2

Question 12

(3^1/4)^n=3^x/81^y

Express n in terms of x and y.

Answer 12

3^n/4=3^x/81^y
3^n/4=3^x/3^4y
3^n/4=3^x-4y
n/4=x-4y
n=4(x-4y)
n=4x-16y

Question 13

Simplify 2x^2-x
x^5

Answer 13

In these questions, split:
2x^2/x^5 -x/x^5

=2/x^3 - 1/x^4
=2x^-3-x^-4

Question 14

As y=3^x,

Express the following in terms of y
1/27^5x-2

Answer 14

(y=3^x)
1/3^3(5x-2)
1/3^15x-6
3^-(15x-6)
3^-15x+6

=3^-15x x 3^6
=3^-15x x 729

(y=36x)
729y^15

Question 15

2^x x 4^y=1/2√2

Express y as a function of x

Answer 15

2^x x 2^2y=1/2√2
2^x+2y=1/2√2
2^x+2y=1/2^3/2
2^x+2y=2^-3/2

x+2y=-3/2
2(x+2y)=-3
2x+4y=-3
4y=-3-2x
y=-3/4-2/4x
y=-3/4-1/2x

Question 16

√a x √b=

Answer 16

√ab

Question 17

√a x b=

Answer 17

b√a

Question 18

√a/√b=

Answer 18

√(a/b)

Question 19

With a denominator of √2, how do you rationalise?

Answer 19

x by √2

Question 20

Denominator of √2-1?

Answer 20

x by √2+1

Question 21

Denominator of 5√2-7

Answer 21

x by 5√2 + 7

Question 22

First step for y(√3-1)=8

Answer 22

y=8/√3-1

Rationalise …

Question 23

Express (x+√x)2
2√x

as powers of x

Answer 23

First, expand

x^2+2√x+x
2√x

Next. split:

x^2/2x^1/2 +2√x/2x^1/2 +x/2x^1/2

Then simplify:
x^3/2/2 + (2x^3/2/2x^1/2=x) x + (x/2x^1/2=x^1/2/2)x^1/2/2

Question 24

Rationalise 1/1+1/√2

Answer 24

Times by √2
√2/√2+1

Times by √2-1

….