Surds and Rationalisation Practice Questions — JAMB UTME

What is the square root of the surd expression $14 + 6\sqrt{5}$ ?

3 + \sqrt{5}

3 - \sqrt{5}

\sqrt{14} + \sqrt{5}

9 + \sqrt{5}

Show answer & explanation

Answer: $3 + \sqrt{5}$

Answer

3 + \sqrt{5}

Expanding the square of

3 + \sqrt{5}

yields

(3)^2 + 2(3)(\sqrt{5}) + (\sqrt{5})^2 = 9 + 6\sqrt{5} + 5 = 14 + 6\sqrt{5}

, which accurately equals the original expression under the radical.

Step-by-Step Solution

1

Set up the general form for the square root of a binomial surd

Let

\sqrt{14 + 6\sqrt{5}} = \sqrt{a} + \sqrt{b}

The square root of a compound surd expression takes the form of a sum of radical terms

2

Square both sides of the equation

14 + 6\sqrt{5} = a + b + 2\sqrt{ab}

Eliminate the outer radical to equate real and surd parts

3

Equate the rational parts and the surd parts

a + b = 14

and

2\sqrt{ab} = 6\sqrt{5} \implies \sqrt{ab} = 3\sqrt{5} = \sqrt{45} \implies ab = 45

Match integer terms together and radical terms together

4

Solve for values of

a

and

b

Two positive numbers with sum

14

and product

45

are

9

and

5

, so

a = 9

and

b = 5

Determine the factors satisfying both equations

5

Substitute

a

and

b

into the radical expression

\sqrt{14 + 6\sqrt{5}} = \sqrt{9} + \sqrt{5} = 3 + \sqrt{5}

Simplify

\sqrt{9}

to

3

to obtain the final simplified expression

Key Concept

Finding the square root of a surd expression by equating rational and radical parts

Question 2Question →

If $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} - \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = x\sqrt{15}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is 2.

Rationalising both fractions yields

4 + \sqrt{15}

and

4 - \sqrt{15}

. Subtracting the second from the first gives

(4 + \sqrt{15}) - (4 - \sqrt{15}) = 2\sqrt{15}

. Comparing

2\sqrt{15}

with

x\sqrt{15}

gives

x = 2

.

Step-by-Step Solution

1

Rationalise the denominator of the first fraction

\frac{(\sqrt{5}+\sqrt{3})^2}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})} = \frac{5 + 2\sqrt{15} + 3}{5 - 3} = 4 + \sqrt{15}

Multiplying numerator and denominator by the conjugate of the denominator removes the surd from the denominator.

2

Rationalise the denominator of the second fraction

\frac{(\sqrt{5}-\sqrt{3})^2}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} = \frac{5 - 2\sqrt{15} + 3}{5 - 3} = 4 - \sqrt{15}

Multiply by the conjugate

(\sqrt{5}-\sqrt{3})

to simplify the second surd term.

3

Subtract the simplified expressions

(4 + \sqrt{15}) - (4 - \sqrt{15}) = 4 - 4 + \sqrt{15} + \sqrt{15} = 2\sqrt{15}

Distribute the negative sign and combine like surd terms.

4

Solve for the unknown coefficient x

2\sqrt{15} = x\sqrt{15} \implies x = 2

Divide both sides of the equation by

\sqrt{15}

to isolate

x

.

Key Concept

Binomial Surd Rationalisation and Simplification

Estimated Time:1m 30s

Question 3Question →

If $\sqrt{x + 4} - \sqrt{x - 1} = 1$ , what is the value of $x$ ?

5

17

3

10

Show answer & explanation

Answer: $5$

Answer

The value of

x

is

5

.

Isolating

\sqrt{x + 4}

gives

\sqrt{x + 4} = 1 + \sqrt{x - 1}

. Squaring both sides yields

x + 4 = 1 + 2\sqrt{x - 1} + x - 1

, which simplifies to

4 = 2\sqrt{x - 1}

. Dividing by 2 gives

2 = \sqrt{x - 1}

. Squaring both sides once more gives

4 = x - 1

, which leads to

x = 5

.

Step-by-Step Solution

1

Isolate one of the surd terms on the left side of the equation.

\sqrt{x + 4} = 1 + \sqrt{x - 1}

Rearranging terms prevents dealing with complex cross-products when squaring.

2

Square both sides of the equation.

x + 4 = 1 + 2\sqrt{x - 1} + (x - 1)

Squaring eliminates the outer radical on the left side.

3

Simplify both sides and isolate the remaining radical term.

4 = 2\sqrt{x - 1} \implies 2 = \sqrt{x - 1}

Subtracting

x

from both sides simplifies the algebraic expression.

4

Square both sides again to solve for

x

.

4 = x - 1 \implies x = 5

Squaring eliminates the remaining radical term.

Key Concept

Solving Surd Equations by Rational Elimination

Question 4Question →

If $\sqrt{75} + \sqrt{12} - \sqrt{27} = k\sqrt{3}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is 4.

Simplifying each surd into its basic form yields

\sqrt{75} = 5\sqrt{3}

,

\sqrt{12} = 2\sqrt{3}

, and

\sqrt{27} = 3\sqrt{3}

. Combining the coefficients gives

(5 + 2 - 3)\sqrt{3} = 4\sqrt{3}

. Equating

4\sqrt{3}

to

k\sqrt{3}

shows that

k = 4

.

Step-by-Step Solution

1

Simplify each individual surd by factoring out perfect squares

\sqrt{75} = 5\sqrt{3}

,

\sqrt{12} = 2\sqrt{3}

,

\sqrt{27} = 3\sqrt{3}

To combine surds through addition or subtraction, they must be converted to similar surds.

2

Combine the coefficients of the like surds

5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} = (5 + 2 - 3)\sqrt{3} = 4\sqrt{3}

Like terms with the same radical factor

\sqrt{3}

can be added and subtracted directly.

3

Compare the resulting coefficient with

k\sqrt{3}

k = 4

By direct comparison of coefficients of

\sqrt{3}

,

k

equals 4.

Key Concept

Simplification and combining of similar surds

Question 5Question →

If $\frac{4 + 3\sqrt{2}}{3 - 2\sqrt{2}}$ is expressed in the simplified form $a + b\sqrt{2}$ , where $a$ and $b$ are integers, what is the value of $a + b$ ?

Show answer & explanation

Answer: 41

Answer

The value of

a + b

is 41.

To rationalise

\frac{4 + 3\sqrt{2}}{3 - 2\sqrt{2}}

, multiply both numerator and denominator by the conjugate

3 + 2\sqrt{2}

. The denominator becomes

3^2 - (2\sqrt{2})^2 = 9 - 8 = 1

. Expanding the numerator gives

(4)(3) + 4(2\sqrt{2}) + 3\sqrt{2}(3) + 3\sqrt{2}(2\sqrt{2}) = 12 + 8\sqrt{2} + 9\sqrt{2} + 12 = 24 + 17\sqrt{2}

. Comparing with

a + b\sqrt{2}

gives

a = 24

and

b = 17

, so

a + b = 24 + 17 = 41

.

Step-by-Step Solution

1

Multiply the numerator and denominator by the conjugate of the denominator.

The expression becomes

\frac{(4 + 3\sqrt{2})(3 + 2\sqrt{2})}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})}

.

Multiplying by the conjugate eliminates surds from the denominator using the difference of two squares identity

(x-y)(x+y) = x^2 - y^2

.

2

Simplify the denominator.

(3)^2 - (2\sqrt{2})^2 = 9 - (4 \times 2) = 9 - 8 = 1

.

Squaring

2\sqrt{2}

yields

2^2 \times (\sqrt{2})^2 = 4 \times 2 = 8

.

3

Expand the numerator.

(4 \times 3) + (4 \times 2\sqrt{2}) + (3\sqrt{2} \times 3) + (3\sqrt{2} \times 2\sqrt{2}) = 12 + 8\sqrt{2} + 9\sqrt{2} + 12 = 24 + 17\sqrt{2}

.

Applying the distributive law and grouping rational terms together and like surd terms together.

4

Identify the values of

a

and

b

and calculate

a + b

.

a = 24

,

b = 17

, so

a + b = 24 + 17 = 41

.

Matching coefficients of the simplified surd form

a + b\sqrt{2}

.

Key Concept

Rationalisation of binomial surd denominators using conjugates

Estimated Time:2m 0s

Question 6Question →

If $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}$ is expressed in the form $a + b\sqrt{3}$ , where $a$ and $b$ are integers, what is the value of $a + b$ ?

3

2

5

1

Show answer & explanation

Answer: 3

Answer

The value of

a + b

is 3.

Multiplying both numerator and denominator by the conjugate

(\sqrt{6} + \sqrt{2})

yields

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

. Comparing this to

a + b\sqrt{3}

gives

a = 2

and

b = 1

, so

a + b = 3

.

Step-by-Step Solution

1

Rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator,

(\sqrt{6} + \sqrt{2})

.

\frac{(\sqrt{6} + \sqrt{2})(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})}

Multiplying by the conjugate eliminates radicals from the denominator using the difference of squares.

2

Expand the numerator

(\sqrt{6} + \sqrt{2})^2

using

(x+y)^2 = x^2 + 2xy + y^2

.

(\sqrt{6})^2 + 2\sqrt{12} + (\sqrt{2})^2 = 6 + 2(2\sqrt{3}) + 2 = 8 + 4\sqrt{3}

Simplifying

\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

converts compound surds to standard form.

3

Expand the denominator using the difference of two squares

(x-y)(x+y) = x^2 - y^2

.

(\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4

Squaring each square root leaves rational integers in the denominator.

4

Divide the expanded numerator by the denominator to find

a

and

b

.

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

Dividing each term by 4 gives

a = 2

and

b = 1

.

5

Compute the sum

a + b

.

2 + 1 = 3

Adding the coefficients

a

and

b

yields the required value.

Key Concept

Rationalisation of Binomial Denominators

Question 7Question →

Simplify the expression $\frac{\sqrt{3}}{\sqrt{3} - 1} - \frac{\sqrt{3}}{\sqrt{3} + 1}$ .

\sqrt{3}

2\sqrt{3}

3

0

Show answer & explanation

Answer: $\sqrt{3}$

Answer

\sqrt{3}

Combining the fractions using the common denominator

(\sqrt{3}-1)(\sqrt{3}+1) = 2

gives a numerator of

\sqrt{3}(\sqrt{3}+1) - \sqrt{3}(\sqrt{3}-1) = 3 + \sqrt{3} - 3 + \sqrt{3} = 2\sqrt{3}

. Dividing

2\sqrt{3}

by

2

yields the simplified answer

\sqrt{3}

.

Step-by-Step Solution

1

Find a common denominator for the two fractions

The common denominator is

(\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2

The denominators are conjugate surds, so their product simplifies to a rational number using the difference of two squares.

2

Express the numerator over the common denominator

Numerator

= \sqrt{3}(\sqrt{3} + 1) - \sqrt{3}(\sqrt{3} - 1)

Multiply each numerator by the missing factor of the common denominator.

3

Expand and simplify the numerator

Numerator

= (3 + \sqrt{3}) - (3 - \sqrt{3}) = 3 + \sqrt{3} - 3 + \sqrt{3} = 2\sqrt{3}

Distribute

\sqrt{3}

and handle the subtraction sign carefully.

4

Divide the simplified numerator by the common denominator

\frac{2\sqrt{3}}{2} = \sqrt{3}

Cancel the common factor of 2.

Key Concept

Rationalisation of denominators and algebraic manipulation of surd fractions

Estimated Time:1m 15s

Question 8Question →

If $\frac{7}{3\sqrt{2} - \sqrt{11}} = a\sqrt{2} + b\sqrt{11}$ , where $a$ and $b$ are rational numbers, what is the value of $a + b$ ?

4

2

12

8

Show answer & explanation

Answer: 4

Answer

4

Multiplying both the top and bottom by the conjugate

(3\sqrt{2} + \sqrt{11})

transforms the denominator into

(3\sqrt{2})^2 - (\sqrt{11})^2 = 18 - 11 = 7

. Dividing the numerator

7(3\sqrt{2} + \sqrt{11})

by

7

simplifies to

3\sqrt{2} + \sqrt{11}

. Matching coefficients yields

a = 3

and

b = 1

, giving

a + b = 4

.

Step-by-Step Solution

1

Multiply the numerator and denominator by the conjugate of the denominator,

(3\sqrt{2} + \sqrt{11})

.

\frac{7(3\sqrt{2} + \sqrt{11})}{(3\sqrt{2} - \sqrt{11})(3\sqrt{2} + \sqrt{11})}

Rationalising the denominator eliminates radicals from the bottom of the fraction.

2

Expand the denominator using the difference of squares formula

(x - y)(x + y) = x^2 - y^2

.

(3\sqrt{2})^2 - (\sqrt{11})^2 = (9 \times 2) - 11 = 18 - 11 = 7

Squaring each term simplifies the denominator into an integer.

3

Simplify the overall rational fraction by cancelling common factors.

\frac{7(3\sqrt{2} + \sqrt{11})}{7} = 3\sqrt{2} + \sqrt{11}

The factor of 7 in the numerator and denominator cancels out.

4

Equate

3\sqrt{2} + 1\sqrt{11}

with

a\sqrt{2} + b\sqrt{11}

to determine the values of

a

and

b

, then sum them.

a = 3, b = 1 \implies a + b = 3 + 1 = 4

Comparing coefficients of corresponding surd terms gives the required values.

Key Concept

Rationalisation of binomial surd denominators using conjugates

Question 9Question →

If $\frac{\sqrt{50} + \sqrt{18}}{\sqrt{2}} = m$ , what is the value of the integer $m$ ?

Show answer & explanation

Answer: 8

Answer

The value of the integer

m

is 8.

Simplifying

\sqrt{50}

to

5\sqrt{2}

and

\sqrt{18}

to

3\sqrt{2}

gives a numerator of

8\sqrt{2}

. Dividing

8\sqrt{2}

by

\sqrt{2}

cancels out the radical part, yielding the integer 8.

Step-by-Step Solution

1

Simplify the radical expressions in the numerator.

\sqrt{50} = 5\sqrt{2}

and

\sqrt{18} = 3\sqrt{2}

.

Factor out perfect square numbers from within each radical.

2

Sum the simplified surds in the numerator.

5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}

.

Surds with identical radicands are like terms and can be added by adding their coefficients.

3

Divide the numerator by the denominator.

\frac{8\sqrt{2}}{\sqrt{2}} = 8

.

Cancel the common factor of

\sqrt{2}

present in both numerator and denominator.

Key Concept

Simplification and division of surds

Question 10Question →

What is the simplified form of the expression $\frac{12}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$ ?

3\sqrt{2} + 2\sqrt{3} - \sqrt{30}

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

2\sqrt{2} + 3\sqrt{3} + \sqrt{30}

3\sqrt{2} + 2\sqrt{3} + \sqrt{10}

Show answer & explanation

Answer: $3\sqrt{2} + 2\sqrt{3} + \sqrt{30}$

Answer

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

By grouping the denominator as

(\sqrt{2}+\sqrt{3}) - \sqrt{5}

and multiplying by its conjugate

(\sqrt{2}+\sqrt{3}) + \sqrt{5}

, the denominator reduces to

2\sqrt{6}

. Multiplying the resulting fraction by

\sqrt{6}/\sqrt{6}

yields

\sqrt{12} + \sqrt{18} + \sqrt{30}

, which simplifies directly to

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

.

Step-by-Step Solution

1

Group the terms in the denominator as

((\sqrt{2} + \sqrt{3}) - \sqrt{5})

and multiply the numerator and denominator by its conjugate

((\sqrt{2} + \sqrt{3}) + \sqrt{5})

.

The fraction becomes

\frac{12((\sqrt{2} + \sqrt{3}) + \sqrt{5})}{((\sqrt{2} + \sqrt{3}) - \sqrt{5})((\sqrt{2} + \sqrt{3}) + \sqrt{5})}

.

Applying the difference of two squares to eliminate the outer radical.

2

Expand the denominator using

(a-b)(a+b) = a^2 - b^2

.

Denominator =

(\sqrt{2} + \sqrt{3})^2 - (\sqrt{5})^2 = (2 + 2\sqrt{6} + 3) - 5 = 2\sqrt{6}

.

Simplifying the algebraic square of a binomial surd.

3

Divide the numerator by the constant factor of the denominator.

\frac{12(\sqrt{2} + \sqrt{3} + \sqrt{5})}{2\sqrt{6}} = \frac{6(\sqrt{2} + \sqrt{3} + \sqrt{5})}{\sqrt{6}}.

Simplifying numerical coefficients before further rationalization.

4

Rationalize the remaining monomial radical in the denominator by multiplying numerator and denominator by

\sqrt{6}

.

\frac{6(\sqrt{12} + \sqrt{18} + \sqrt{30})}{6} = \sqrt{12} + \sqrt{18} + \sqrt{30}.

Eliminating

\sqrt{6}

from the denominator.

5

Simplify each radical to its simplest surd form.

\sqrt{12} = 2\sqrt{3}, \quad \sqrt{18} = 3\sqrt{2}, \quad \text{so } \sqrt{12} + \sqrt{18} + \sqrt{30} = 3\sqrt{2} + 2\sqrt{3} + \sqrt{30}.

Factoring out perfect square components from radical terms.

Key Concept

Rationalisation of trinomial surd denominators using repeated conjugate multiplication

Estimated Time:2m 0s

Question 11Question →

If $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} - \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} = k\sqrt{6}$ , find the value of the integer $k$ .

Show answer & explanation

Answer: 4

Answer

The value of the integer

k

is 4.

Combining the fractions over the common denominator

(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}) = 3-2 = 1

yields a numerator of

(3+2+2\sqrt{6}) - (3+2-2\sqrt{6}) = 4\sqrt{6}

. Thus,

k\sqrt{6} = 4\sqrt{6}

, which gives

k = 4

.

Step-by-Step Solution

1

Combine the fractions using their common denominator

\frac{(\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}

Subtracting algebraic fractions requires finding the least common denominator, which is the product of the conjugate pair.

2

Expand the terms in the numerator

(\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \text{ and } (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6}

Use the perfect square expansion formula

(a \pm b)^2 = a^2 \pm 2ab + b^2

.

3

Subtract the expanded terms in the numerator and simplify the denominator

\text{Numerator: } (5 + 2\sqrt{6}) - (5 - 2\sqrt{6}) = 4\sqrt{6}, \text{ Denominator: } (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1

Apply the difference of two squares identity

(a-b)(a+b) = a^2 - b^2

to the denominator and carefully distribute the negative sign across terms in the numerator.

4

Equate the simplified expression to

k\sqrt{6}

and solve for

k

k = 4

Comparing

\frac{4\sqrt{6}}{1} = 4\sqrt{6}

with

k\sqrt{6}

yields

k = 4

.

Key Concept

Rationalisation of surd denominators using conjugate pairs and difference of squares

Question 12Question →

If $\sqrt{28 - 10\sqrt{3}} + \sqrt{7 + 4\sqrt{3}} = k$ , where $k$ is a rational number, find the value of $k$ .

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

Simplifying each nested surd into standard binomial surd form gives

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

and

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

. Combining these terms cancels out the irrational component

\sqrt{3}

, leaving the integer result

5 + 2 = 7

.

Step-by-Step Solution

1

Simplify the nested surd

\sqrt{28 - 10\sqrt{3}}

into binomial surd form

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

Assuming a form

x - y\sqrt{3}

and squaring both sides gives

x^2 + 3y^2 - 2xy\sqrt{3} = 28 - 10\sqrt{3}

. Matching components leads to

xy = 5

and

x^2 + 3y^2 = 28

, which yields integer values

x = 5

and

y = 1

.

2

Simplify the nested surd

\sqrt{7 + 4\sqrt{3}}

into binomial surd form

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

Assuming a form

u + v\sqrt{3}

and squaring both sides gives

u^2 + 3v^2 + 2uv\sqrt{3} = 7 + 4\sqrt{3}

. Matching components leads to

uv = 2

and

u^2 + 3v^2 = 7

, which yields integer values

u = 2

and

v = 1

.

3

Sum the simplified expressions to calculate

k

k = 7

Summing

(5 - \sqrt{3}) + (2 + \sqrt{3})

results in the irrational parts

-\sqrt{3}

and

\sqrt{3}

cancelling out, leaving

5 + 2 = 7

.

Key Concept

Square Root of a Binomial Surd Expression

Question 13Question →

If $\frac{3\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = a + b\sqrt{15}$ , where $a$ and $b$ are rational numbers, what is the value of $a - b$ ?

11

8

5

13

Show answer & explanation

Answer: 11

Answer

11

By multiplying the numerator and denominator by the conjugate of the denominator,

(\sqrt{5} - \sqrt{3})

, the fraction simplifies to

\frac{18 - 4\sqrt{15}}{2} = 9 - 2\sqrt{15}

. Equating this to

a + b\sqrt{15}

gives

a = 9

and

b = -2

. Calculating

a - b

gives

9 - (-2) = 11

.

Step-by-Step Solution

1

Multiply the numerator and denominator by the conjugate of the denominator

\frac{3\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{(3\sqrt{5} - \sqrt{3})(\sqrt{5} - \sqrt{3})}{(\sqrt{5})^2 - (\sqrt{3})^2}

Rationalising the denominator requires using the difference of squares identity

(x+y)(x-y) = x^2 - y^2

.

2

Expand the numerator and denominator

Denominator = 5 - 3 = 2. Numerator = 3(5) - 3\sqrt{15} - \sqrt{15} + 3 = 15 + 3 - 4\sqrt{15} = 18 - 4\sqrt{15}.

Apply distributive property to expand

(3\sqrt{5} - \sqrt{3})(\sqrt{5} - \sqrt{3})

carefully combining like terms.

3

Simplify the fraction to match the form

a + b\sqrt{15}

\frac{18 - 4\sqrt{15}}{2} = 9 - 2\sqrt{15}

Divide each term in the numerator by 2.

4

Identify

a

and

b

and evaluate

a - b

a = 9, b = -2 \implies a - b = 9 - (-2) = 11

Subtracting negative 2 is equivalent to adding 2.

Key Concept

Rationalisation of Binomial Surd Denominators

Estimated Time:1m 30s

Question 14Question →

If $\frac{2 + \sqrt{3}}{2 - \sqrt{3}} = x + y\sqrt{3}$ , where $x$ and $y$ are integers, what is the value of $x + y$ ?

Show answer & explanation

Answer: 11

Answer

The value of

x + y

is 11.

Multiplying the given fraction by

\frac{2 + \sqrt{3}}{2 + \sqrt{3}}

rationalises the denominator to 1 and simplifies the numerator to

7 + 4\sqrt{3}

. Comparing coefficients yields

x = 7

and

y = 4

, making

x + y = 11

.

Step-by-Step Solution

1

Multiply numerator and denominator by the conjugate of the denominator

\frac{(2 + \sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}

Rationalising the denominator eliminates the surd from the denominator using the identity (a-b)(a+b) = a^2 - b^2.

2

Expand both the numerator and the denominator

\frac{4 + 4\sqrt{3} + 3}{4 - 3} = \frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3}

Simplifying algebraic surd multiplication gives integer and surd terms.

3

Compare terms with x + y\sqrt{3} and solve for x and y

x = 7, y = 4 \implies x + y = 11

Matching rational components and coefficients of \sqrt{3} yields x and y.

Key Concept

Rationalisation of Binomial Denominators containing Surds

Estimated Time:1m 30s

Question 15Question →

If $\frac{4\sqrt{3}}{\sqrt{7} + \sqrt{3}} + \frac{4\sqrt{7}}{\sqrt{7} - \sqrt{3}} = p + q\sqrt{21}$ , where $p$ and $q$ are integers, what is the value of $p + q$ ?

Show answer & explanation

Answer: 6

Answer

The value of

p + q

is 6.

Rationalising each fraction yields

(\sqrt{21} - 3)

and

(7 + \sqrt{21})

. Summing these expressions gives

4 + 2\sqrt{21}

. Comparing this to

p + q\sqrt{21}

yields

p = 4

and

q = 2

, so

p + q = 6

.

Step-by-Step Solution

1

Rationalise the first term

\frac{4\sqrt{3}}{\sqrt{7} + \sqrt{3}}

by multiplying the numerator and denominator by the conjugate

(\sqrt{7} - \sqrt{3})

.

\frac{4\sqrt{3}(\sqrt{7} - \sqrt{3})}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{4\sqrt{21} - 12}{7 - 3} = \frac{4\sqrt{21} - 12}{4} = \sqrt{21} - 3

Multiplying by the conjugate eliminates the surd from the denominator using the difference of two squares.

2

Rationalise the second term

\frac{4\sqrt{7}}{\sqrt{7} - \sqrt{3}}

by multiplying the numerator and denominator by the conjugate

(\sqrt{7} + \sqrt{3})

.

\frac{4\sqrt{7}(\sqrt{7} + \sqrt{3})}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{28 + 4\sqrt{21}}{7 - 3} = \frac{28 + 4\sqrt{21}}{4} = 7 + \sqrt{21}

Conjugate rationalisation simplifies the second fraction into linear surd terms.

3

Add the two simplified expressions together and equate to

p + q\sqrt{21}

.

(\sqrt{21} - 3) + (7 + \sqrt{21}) = 4 + 2\sqrt{21}

Combining like surd terms yields the simplified form

p + q\sqrt{21}

.

4

Identify the values of

p

and

q

and evaluate

p + q

.

p = 4, q = 2 \implies p + q = 4 + 2 = 6

Equating the rational parts gives

p = 4

and the irrational coefficients gives

q = 2

.

Key Concept

Rationalisation of surds with binomial denominators

Alternative Method

Combine the two fractions directly over the common denominator

(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = 4

: \frac{4\sqrt{3}(\sqrt{7} - \sqrt{3}) + 4\sqrt{7}(\sqrt{7} + \sqrt{3})}{4} = \frac{4\sqrt{21} - 12 + 28 + 4\sqrt{21}}{4} = \frac{16 + 8\sqrt{21}}{4} = 4 + 2\sqrt{21}.

Estimated Time:1m 30s

Question 16Question →

What is the simplified value of $\frac{6}{\sqrt{7} - 1} - \frac{6}{\sqrt{7} + 1}$ ?

2

2\sqrt{7}

12

12\sqrt{7}

Show answer & explanation

Answer: 2

Answer

2

Combining the fractions over the common denominator

(\sqrt{7} - 1)(\sqrt{7} + 1) = 6

yields a numerator of

6(\sqrt{7} + 1) - 6(\sqrt{7} - 1) = 12

. Dividing 12 by 6 gives 2.

Step-by-Step Solution

1

Find a common denominator for the two fractions

The common denominator is

(\sqrt{7} - 1)(\sqrt{7} + 1) = (\sqrt{7})^2 - 1^2 = 7 - 1 = 6

.

Multiplying conjugate surds eliminates the radical in the denominator.

2

Combine the numerators over the common denominator

\frac{6(\sqrt{7} + 1) - 6(\sqrt{7} - 1)}{6}

Adjust each numerator by multiplying by the conjugate of its denominator.

3

Expand and simplify the numerator

6\sqrt{7} + 6 - 6\sqrt{7} + 6 = 12

The

6\sqrt{7}

terms cancel out:

6\sqrt{7} - 6\sqrt{7} = 0

, leaving

6 - (-6) = 12

.

4

Divide the simplified numerator by the denominator

\frac{12}{6} = 2

Simplify the final fraction to obtain an integer value.

Key Concept

Rationalisation and Subtraction of Surd Expressions

Estimated Time:1m 0s

Surds and Rationalisation

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