Surds and Rationalization of Denominators Practice Questions — JAMB UTME

What value of $x$ satisfies the surd equation $\sqrt{x + 7} - \sqrt{x} = 1$ ?

9

3

16

7

Show answer & explanation

Answer: $9$

Answer

The value of

x

that satisfies the equation is

9

.

By rearranging the equation to

\sqrt{x+7} = \sqrt{x} + 1

and squaring both sides, we get

x + 7 = x + 2\sqrt{x} + 1

. Subtracting

x + 1

from both sides gives

6 = 2\sqrt{x}

, which yields

\sqrt{x} = 3

. Squaring both sides produces

x = 9

, which correctly satisfies the original equation.

Step-by-Step Solution

1

Isolate one of the radical terms on one side of the equation.

\sqrt{x + 7} = \sqrt{x} + 1

Isolating a square root allows squaring both sides to eliminate the outer radical.

2

Square both sides of the equation.

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

Expanding the right-hand side using

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

removes the radical from the left side.

3

Subtract

x

and

1

from both sides to isolate the remaining radical term.

6 = 2\sqrt{x} \implies \sqrt{x} = 3

Simplifying the linear terms leaves a simple square root equation.

4

Square both sides to find

x

.

x = 3^2 = 9

Squaring

\sqrt{x}

isolates

x

completely.

Key Concept

Solving Surd Equations by Isolating Radicals and Squaring

Question 2Question →

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

\frac{5 + \sqrt{15}}{2}

\sqrt{5}

5 + \sqrt{15}

\frac{5 - \sqrt{15}}{2}

Show answer & explanation

Answer: $\frac{5 + \sqrt{15}}{2}$

Answer

\frac{5 + \sqrt{15}}{2}

To rationalize the denominator of

\frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}}

, multiply both numerator and denominator by the conjugate

\sqrt{5} + \sqrt{3}

. The numerator becomes

\sqrt{5}(\sqrt{5} + \sqrt{3}) = 5 + \sqrt{15}

and the denominator becomes

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

, yielding

\frac{5 + \sqrt{15}}{2}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

\sqrt{5} - \sqrt{3}

is

\sqrt{5} + \sqrt{3}

.

Multiplying a binomial surd by its conjugate eliminates the radical terms in the denominator using the difference of two squares identity

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

.

2

Multiply both the numerator and the denominator by the conjugate

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

Multiplying the fraction by

\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}

is equivalent to multiplying by

1

, preserving the value of the expression.

3

Expand the numerator and simplify the denominator

Numerator:

\sqrt{5} \times \sqrt{5} + \sqrt{5} \times \sqrt{3} = 5 + \sqrt{15}

. Denominator:

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

.

Applying the distributive law to the numerator and difference of squares to the denominator simplifies both parts.

4

Combine the simplified numerator and denominator

\frac{5 + \sqrt{15}}{2}

This is the simplified rationalized form.

Key Concept

Rationalization of Denominators

Question 3Question →

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

5

7

9

13

Show answer & explanation

Answer: 7

Answer

The correct value of

a + b

is 7.

Multiplying by the conjugate

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

reduces the denominator to

18 - 12 = 6

and expands the numerator to

30 + 12\sqrt{6}

. Dividing by 6 yields

5 + 2\sqrt{6}

, giving

a = 5

and

b = 2

, which sums to 7.

Step-by-Step Solution

1

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

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

.

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

Rationalizing eliminates the surd terms from the denominator using the difference of two squares.

2

Expand both numerator and denominator.

Denominator:

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

. Numerator:

(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}

.

Apply algebraic identities

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

and

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

.

3

Divide the numerator by the denominator to simplify the expression.

\frac{30 + 12\sqrt{6}}{6} = 5 + 2\sqrt{6}

Both integer and radical coefficients are divisible by 6.

4

Equate

5 + 2\sqrt{6}

to

a + b\sqrt{6}

and calculate

a + b

.

a = 5, b = 2, so a + b = 5 + 2 = 7.

Matching corresponding rational and irrational components.

Key Concept

Rationalization of Binomial Denominators with Surds

Question 4Question →

What is the numerical value of the simplified expression $\frac{4}{\sqrt{5} - 1} - \sqrt{5}$ ?

Show answer & explanation

Answer: 1

Answer

The numerical value of the expression is 1.

Multiplying the top and bottom of

\frac{4}{\sqrt{5} - 1}

by its conjugate

(\sqrt{5} + 1)

simplifies the fraction to

\frac{4(\sqrt{5} + 1)}{4} = \sqrt{5} + 1

. Subtracting

\sqrt{5}

from

\sqrt{5} + 1

leaves 1.

Step-by-Step Solution

1

Rationalize the denominator of the fractional term

\sqrt{5} + 1

Multiply both numerator and denominator by the conjugate

(\sqrt{5} + 1)

to apply the difference of two squares identity

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

in the denominator.

2

Subtract the remaining surd term

1

Subtract

\sqrt{5}

from

\sqrt{5} + 1

, leaving the integer 1.

Key Concept

Rationalization of Binomial Denominators

Question 5Question →

If the expression $\frac{3 - \sqrt{5}}{3 + \sqrt{5}}$ is simplified and written in the form $a + b\sqrt{5}$ , where $a$ and $b$ are rational numbers, what is the numerical value of $a + b$ ?

Show answer & explanation

Answer: 2

Answer

The numerical value of

a + b

is

2

.

To express

\frac{3 - \sqrt{5}}{3 + \sqrt{5}}

in the standard form

a + b\sqrt{5}

, multiply both the numerator and denominator by the conjugate of the denominator, which is

(3 - \sqrt{5})

. The numerator expands to

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

. The denominator becomes

3^2 - (\sqrt{5})^2 = 9 - 5 = 4

. Dividing gives

\frac{14}{4} - \frac{6}{4}\sqrt{5} = \frac{7}{2} - \frac{3}{2}\sqrt{5}

. Hence,

a = \frac{7}{2}

and

b = -\frac{3}{2}

, making

a + b = \frac{7}{2} - \frac{3}{2} = \frac{4}{2} = 2

.

Step-by-Step Solution

1

Multiply numerator and denominator by the conjugate of the denominator

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

To eliminate the surd from the denominator.

2

Expand both the numerator and the denominator

\frac{14 - 6\sqrt{5}}{4}

Using

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

for the numerator and difference of two squares

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

for the denominator.

3

Separate into rational component and radical coefficient

\frac{7}{2} - \frac{3}{2}\sqrt{5}

Simplifying fractions by dividing numerator and denominator by their greatest common divisor.

4

Calculate the sum

a + b

\frac{7}{2} - \frac{3}{2} = 2

Comparing

\frac{7}{2} - \frac{3}{2}\sqrt{5}

with

a + b\sqrt{5}

yields

a = \frac{7}{2}

and

b = -\frac{3}{2}

.

Key Concept

Rationalization of Binomial Denominators using Conjugates

Estimated Time:1m 30s

Question 6Question →

What is the simplified form of the expression $\sqrt{50} - \sqrt{18} + \sqrt{8}$ ?

4\sqrt{2}

2\sqrt{10}

6\sqrt{2}

2\sqrt{2}

Show answer & explanation

Answer: $4\sqrt{2}$

Answer

The simplified form of the expression is

4\sqrt{2}

.

Simplifying each radical into basic surd form yields

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

,

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

, and

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

. Combining these like terms gives

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

.

Step-by-Step Solution

1

Simplify each surd term into basic surd form

\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}

,

\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}

, and

\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}

Converting each surd to have a common radicand allows for addition and subtraction of like terms.

2

Substitute the simplified surds back into the expression and combine like terms

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

Surds with identical radicands can be combined by operating on their coefficients.

Key Concept

Simplification and Addition/Subtraction of Like Surds

Question 7Question →

If the expression $\frac{\sqrt{48}}{\sqrt{3} + \sqrt{2}} + \frac{\sqrt{72}}{\sqrt{3} - \sqrt{2}}$ is simplified into the form $m + n\sqrt{6}$ , where $m$ and $n$ are integers, find the value of $m + n$ .

Show answer & explanation

Answer: 26

Answer

The simplified expression is

24 + 2\sqrt{6}

, giving

m = 24

and

n = 2

, so

m + n = 26

.

Simplifying

\sqrt{48}

to

4\sqrt{3}

and

\sqrt{72}

to

6\sqrt{2}

allows rationalization of each fraction by its conjugate. The first fraction becomes

12 - 4\sqrt{6}

and the second becomes

12 + 6\sqrt{6}

. Adding these expressions results in

24 + 2\sqrt{6}

, so

m = 24

and

n = 2

, giving

m + n = 26

.

Step-by-Step Solution

1

Simplify the radical numerators

\sqrt{48} = 4\sqrt{3}

and

\sqrt{72} = 6\sqrt{2}

Decomposing surds into perfect square factors simplifies subsequent algebraic expansion.

2

Rationalize the first term

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

12 - 4\sqrt{6}

Multiplying the numerator and denominator by the conjugate

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

removes the surd from the denominator using the difference of squares

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

.

3

Rationalize the second term

\frac{6\sqrt{2}}{\sqrt{3} - \sqrt{2}}

12 + 6\sqrt{6}

Multiplying the numerator and denominator by the conjugate

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

yields a rational denominator of

1

.

4

Combine like surd terms

24 + 2\sqrt{6}

Summing the rational components

(12 + 12 = 24)

and combining similar surd terms

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

.

5

Calculate the target sum

m + n

26

Matching coefficients with

m + n\sqrt{6}

gives

m = 24

and

n = 2

, yielding

24 + 2 = 26

.

Key Concept

Rationalization of Binomial Denominators using Conjugates

Question 8Question →

If the surd expression $\sqrt{75} - \sqrt{12} + \frac{6}{\sqrt{3}}$ is simplified to the form $k\sqrt{3}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is 5.

Simplifying

\sqrt{75}

yields

5\sqrt{3}

, simplifying

\sqrt{12}

yields

2\sqrt{3}

, and rationalizing

\frac{6}{\sqrt{3}}

gives

2\sqrt{3}

. Adding these together gives

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

. Equating

5\sqrt{3}

to

k\sqrt{3}

yields

k = 5

.

Step-by-Step Solution

1

Simplify the individual square roots.

\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}

and

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

.

Factoring out perfect square numbers allows surds to be written in basic radical form.

2

Rationalize the fractional surd term

\frac{6}{\sqrt{3}}

.

\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}

.

Multiplying numerator and denominator by

\sqrt{3}

eliminates the radical from the denominator.

3

Combine like surd terms.

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

.

Like surds share the same radical factor and can be added or subtracted algebraically.

4

Equate the simplified expression to

k\sqrt{3}

.

k = 5

.

Comparing coefficients of

\sqrt{3}

reveals the value of

k

.

Key Concept

Surd Simplification and Rationalization of Denominators

Estimated Time:1m 30s

Question 9Question →

What is the simplified form of $\frac{6}{3 + \sqrt{3}}$ after rationalizing the denominator?

3 - \sqrt{3}

3 + \sqrt{3}

\sqrt{3}

3 - 2\sqrt{3}

Show answer & explanation

Answer: $3 - \sqrt{3}$

Answer

3 - \sqrt{3}

Multiplying the numerator and denominator by the conjugate

3 - \sqrt{3}

converts the denominator into

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

. Dividing

6(3 - \sqrt{3})

by

6

yields

3 - \sqrt{3}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

3 + \sqrt{3}

is

3 - \sqrt{3}

.

To eliminate the surd from the denominator, multiply by its conjugate.

2

Multiply the numerator and denominator by the conjugate

\frac{6(3 - \sqrt{3})}{(3 + \sqrt{3})(3 - \sqrt{3})}

Multiplying by

\frac{3 - \sqrt{3}}{3 - \sqrt{3}}

is equivalent to multiplying by 1.

3

Simplify the denominator using difference of two squares

(3)^2 - (\sqrt{3})^2 = 9 - 3 = 6

The product

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

simplifies the denominator to a rational number.

4

Divide the numerator by the denominator

\frac{6(3 - \sqrt{3})}{6} = 3 - \sqrt{3}

Canceling out the common factor of 6 gives the final simplified surd expression.

Key Concept

Rationalization of Binomial Denominators

Question 10Question →

If $\frac{2\sqrt{3} - \sqrt{5}}{\sqrt{3} + \sqrt{5}} = a + b\sqrt{15}$ , where $a$ and $b$ are rational numbers, what is the value of $a - b$ ?

-7

7

-4

-2

Show answer & explanation

Answer: $-7$

Answer

-7

Rationalizing

\frac{2\sqrt{3} - \sqrt{5}}{\sqrt{3} + \sqrt{5}}

by multiplying top and bottom by

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

yields

\frac{11 - 3\sqrt{15}}{-2} = -\frac{11}{2} + \frac{3}{2}\sqrt{15}

. Comparing this with

a + b\sqrt{15}

gives

a = -\frac{11}{2}

and

b = \frac{3}{2}

. Computing

a - b

gives

-\frac{11}{2} - \frac{3}{2} = -7

.

Step-by-Step Solution

1

Multiply the numerator and denominator by the conjugate of the denominator

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

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

Rationalizing the denominator requires using the difference of squares.

2

Expand the numerator and simplify the denominator

Numerator:

2(3) - 2\sqrt{15} - \sqrt{15} + 5 = 11 - 3\sqrt{15}

. Denominator:

3 - 5 = -2

.

Multiply terms using FOIL and replace

(\sqrt{3})^2

with

3

and

(\sqrt{5})^2

with

5

.

3

Divide the numerator by the denominator to express in standard form

a + b\sqrt{15}

\frac{11 - 3\sqrt{15}}{-2} = -\frac{11}{2} + \frac{3}{2}\sqrt{15}

Separate the rational term and the surd coefficient.

4

Equate coefficients to find

a

and

b

, then compute

a - b

a = -\frac{11}{2}, b = \frac{3}{2} \implies a - b = -\frac{11}{2} - \frac{3}{2} = -\frac{14}{2} = -7

Calculate the target expression

a - b

using the derived rational values.

Key Concept

Rationalization of Binomial Denominators and Equating Surd Coefficients

Question 11Question →

What is the simplified form of the surd expression $\frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} - 2\sqrt{3}}$ after rationalizing the denominator?

5 + 2\sqrt{6}

5 + 12\sqrt{6}

5

5\sqrt{6}

Show answer & explanation

Answer: $5 + 2\sqrt{6}$

Answer

5 + 2\sqrt{6}

To rationalize

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

, we multiply both the numerator and denominator by the conjugate of the denominator,

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

. Expanding the numerator yields

(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}

. Expanding the denominator gives

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

. Dividing

30 + 12\sqrt{6}

by

6

gives

5 + 2\sqrt{6}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

3\sqrt{2} - 2\sqrt{3}

is

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

.

Multiplying by the conjugate converts the binomial denominator into a rational number using the difference of squares identity

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

.

2

Multiply the numerator and denominator by the conjugate

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

This maintains expression equivalence while eliminating radicals from the denominator.

3

Expand the numerator and denominator independently

Numerator:

(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}

. Denominator:

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

.

Apply algebraic expansion

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

and simplification of radicals.

4

Divide each term in the numerator by the denominator

\frac{30 + 12\sqrt{6}}{6} = \frac{30}{6} + \frac{12\sqrt{6}}{6} = 5 + 2\sqrt{6}.

Simplify the fraction to express the answer in standard surd form

a + b\sqrt{c}

.

Key Concept

Rationalization of Binomial Denominators

Question 12Question →

What is the simplified form of the expression $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} - \sqrt{3}$ ?

2

2 + 2\sqrt{3}

\sqrt{3} - 1

2\sqrt{3}

Show answer & explanation

Answer: $2$

Answer

The simplified form of the expression is

2

.

Multiplying the numerator and denominator of

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

by the conjugate

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

gives

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

. Subtracting

\sqrt{3}

yields

2

.

Step-by-Step Solution

1

Rationalize the denominator of

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

Multiply numerator and denominator by the conjugate

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

to get

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

.

Rationalizing eliminates the surd from the denominator using the difference of two squares identity.

2

Expand the numerator and simplify the fraction

Numerator:

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

. Denominator:

6 - 2 = 4

. Fraction simplifies to

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

.

Simplifying radical factors (

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

) allows division by the common denominator.

3

Subtract

\sqrt{3}

from the rationalized term

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

.

Subtracting like surd terms cancels out

\sqrt{3}

leaving the rational constant.

Key Concept

Rationalization of Binomial Denominators and Surd Simplification

Question 13Question →

Simplify the surd expression $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ .

\frac{7 + 2\sqrt{10}}{3}

\frac{\sqrt{7} + 2\sqrt{10}}{3}

\frac{7 + \sqrt{10}}{3}

\frac{7 + 2\sqrt{10}}{7}

Show answer & explanation

Answer: $\frac{7 + 2\sqrt{10}}{3}$

Answer

\frac{7 + 2\sqrt{10}}{3}

Multiplying both the numerator and denominator by the conjugate of the denominator,

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

, expands the numerator to

5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}

and simplifies the denominator using the difference of squares to

5 - 2 = 3

, giving

\frac{7 + 2\sqrt{10}}{3}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

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

is

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

.

Rationalizing a binomial denominator requires multiplying by its conjugate to apply the difference of squares identity

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

.

2

Multiply both numerator and denominator by the conjugate

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

This maintains the value of the fraction while removing radical terms from the denominator.

3

Expand the numerator and denominator

Numerator:

(\sqrt{5})^2 + 2\sqrt{5}\sqrt{2} + (\sqrt{2})^2 = 5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}

. Denominator:

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

.

Apply algebraic expansion

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

and difference of squares.

4

Combine terms to form the final simplified expression

\frac{7 + 2\sqrt{10}}{3}

The expression is now fully rationalized and in standard simplified surd form.

Key Concept

Rationalization of Binomial Denominators

Question 14Question →

Given that $\frac{\sqrt{x + 2} + \sqrt{x - 2}}{\sqrt{x + 2} - \sqrt{x - 2}} = 3$ , what is the value of $x$ ?

\frac{10}{3}

6

\frac{5}{2}

\frac{8}{3}

Show answer & explanation

Answer: $\frac{10}{3}$

Answer

The value of

x

is

\frac{10}{3}

.

The correct solution is found by cross-multiplying the equation to get

\sqrt{x + 2} + \sqrt{x - 2} = 3\sqrt{x + 2} - 3\sqrt{x - 2}

. Grouping similar surd terms gives

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

, which simplifies to

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

. Squaring both sides yields

4(x - 2) = x + 2

, simplifying to

4x - 8 = x + 2

, which gives

3x = 10

and

x = \frac{10}{3}

.

Step-by-Step Solution

1

Cross-multiply to clear the denominator

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

Clear the fraction to group like radical terms on opposite sides.

2

Rearrange and combine like terms

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

, which simplifies to

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

Isolate the radical expressions.

3

Square both sides of the equation

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

Eliminate radicals by squaring both sides, ensuring the coefficient

2

is squared to

4

.

4

Solve the linear equation for

x

4x - 8 = x + 2 \implies 3x = 10 \implies x = \frac{10}{3}

Isolate

x

to find the final value.

Key Concept

Solving Radical and Surd Equations

Question 15Question →

What is the simplified form of the surd expression $\frac{4}{\sqrt{5} - 1}$ after rationalizing the denominator?

1 + \sqrt{5}

2

4 + 4\sqrt{5}

\sqrt{5} - 1

Show answer & explanation

Answer: $1 + \sqrt{5}$

Answer

1 + \sqrt{5}

The expression

1 + \sqrt{5}

is correct because multiplying the numerator and denominator by the conjugate

\sqrt{5} + 1

converts the denominator to

(\sqrt{5})^2 - 1^2 = 4

. Canceling the common factor of

4

in numerator and denominator simplifies the expression completely to

1 + \sqrt{5}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

\sqrt{5} - 1

is

\sqrt{5} + 1

.

Multiplying a binomial surd by its conjugate eliminates the radical in the denominator using the difference of two squares identity

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

.

2

Multiply both the numerator and the denominator by the conjugate

\frac{4(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{4(\sqrt{5} + 1)}{(\sqrt{5})^2 - (1)^2}

Multiplying both numerator and denominator by the same expression preserves the value of the fraction.

3

Simplify the denominator and evaluate the fraction

\frac{4(\sqrt{5} + 1)}{5 - 1} = \frac{4(\sqrt{5} + 1)}{4} = 1 + \sqrt{5}

Dividing the numerator by

4

cancels out the factor of

4

.

Key Concept

Rationalization of Binomial Denominators

Question 16Question →

If $x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}$ and $y = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}$ , determine the numerical value of $x^2 + y^2$ .

Show answer & explanation

Answer: 23

Answer

The numerical value of

x^2 + y^2

is

23

.

Rationalizing

x

yields

\frac{5 + \sqrt{21}}{2}

and rationalizing

y

yields

\frac{5 - \sqrt{21}}{2}

. The sum

x + y

equals

5

and the product

xy

equals

1

. Substituting these into

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

gives

5^2 - 2(1) = 23

.

Step-by-Step Solution

1

Rationalize the denominators of

x

and

y

x = \frac{5 + \sqrt{21}}{2}

and

y = \frac{5 - \sqrt{21}}{2}

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

2

Calculate the sum

x + y

and the product

xy

x + y = 5

and

xy = 1

Summing conjugate surd expressions cancels the radical term, and multiplying them applies the difference of two squares.

3

Evaluate

x^2 + y^2

using the identity

(x + y)^2 - 2xy

x^2 + y^2 = 5^2 - 2(1) = 23

Substituting the known sum and product avoids having to square complex surd expressions directly.

Key Concept

Rationalization of binomial denominators and application of symmetric algebraic identities.

Question 17Question →

If the surd expression $\frac{12}{\sqrt{6}}$ is simplified to the form $k\sqrt{6}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The value of

k

is 2.

Multiplying the numerator and denominator of

\frac{12}{\sqrt{6}}

by

\sqrt{6}

yields

\frac{12\sqrt{6}}{6}

. Dividing

12

by

6

simplifies the expression to

2\sqrt{6}

. Matching this with

k\sqrt{6}

gives

k = 2

.

Step-by-Step Solution

1

Rationalize the denominator of

\frac{12}{\sqrt{6}}

by multiplying the numerator and denominator by

\sqrt{6}

.

\frac{12\sqrt{6}}{6}

Multiplying by

\frac{\sqrt{6}}{\sqrt{6}}

removes the radical from the denominator without changing the value of the expression.

2

Divide the integer coefficient in the numerator by the denominator.

2\sqrt{6}

Simplifying

\frac{12}{6}

yields 2.

3

Compare

2\sqrt{6}

with

k\sqrt{6}

to determine the value of

k

.

k = 2

The coefficient of

\sqrt{6}

is 2.

Key Concept

Rationalization of monomial surd denominators

Question 18Question →

Given that $\frac{\sqrt{5} + 2}{\sqrt{5} - 2} = a + b\sqrt{5}$ , where $a$ and $b$ are rational numbers, what is the value of $a + b$ ?

13

11

9

7

Show answer & explanation

Answer: 13

Answer

The value of

a + b

is 13.

To simplify

\frac{\sqrt{5} + 2}{\sqrt{5} - 2}

, multiply both numerator and denominator by the conjugate of the denominator, which is

\sqrt{5} + 2

. Expanding the numerator

(\sqrt{5} + 2)^2

gives

5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5}

. The denominator simplifies to

(\sqrt{5})^2 - 2^2 = 5 - 4 = 1

. Thus, the expression becomes

9 + 4\sqrt{5}

. Comparing this to

a + b\sqrt{5}

gives

a = 9

and

b = 4

. Therefore,

a + b = 13

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The denominator is

\sqrt{5} - 2

, so its conjugate is

\sqrt{5} + 2

.

Multiplying by the conjugate rationalizes the binomial denominator using the difference of two squares.

2

Multiply the numerator and denominator by the conjugate

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

This removes the radical from the denominator.

3

Expand both numerator and denominator

\frac{5 + 4\sqrt{5} + 4}{5 - 4} = \frac{9 + 4\sqrt{5}}{1} = 9 + 4\sqrt{5}

Using

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

for the numerator and

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

for the denominator.

4

Equate to

a + b\sqrt{5}

and solve for

a + b

a = 9

and

b = 4

, so

a + b = 9 + 4 = 13

.

Matching the rational part

a

and coefficient of the surd

b

gives the target sum.

Key Concept

Rationalization of Binomial Denominators

Estimated Time:1m 15s

Question 19Question →

What is the simplified form of the surd expression $\sqrt{45} - \sqrt{20} + \sqrt{80}$ ?

5\sqrt{5}

\sqrt{105}

9\sqrt{5}

3\sqrt{5}

Show answer & explanation

Answer: $5\sqrt{5}$

Answer

The simplified form of the expression is

5\sqrt{5}

.

Each radical is decomposed into a product involving a perfect square:

\sqrt{45} = 3\sqrt{5}

,

\sqrt{20} = 2\sqrt{5}

, and

\sqrt{80} = 4\sqrt{5}

. Combining the coefficients

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

yields

5\sqrt{5}

.

Step-by-Step Solution

1

Simplify each individual surd into basic radical form by finding perfect square factors.

\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}

,

\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}

, and

\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}

.

Expressing surds in terms of identical basic radicals allows like terms to be combined.

2

Substitute the simplified surds back into the original expression and combine like terms.

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

.

Perform addition and subtraction on the coefficients of similar surds.

Key Concept

Simplification of Surds

Question 20Question →

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

Show answer & explanation

Answer: 27

Answer

The value of

x

is 27.

Cross-multiplying yields

\sqrt{x} + \sqrt{3} = 2\sqrt{x} - 2\sqrt{3}

. Rearranging terms gives

\sqrt{x} = 3\sqrt{3}

. Squaring both sides produces

x = 3^2 \times 3 = 27

.

Step-by-Step Solution

1

Multiply both sides by the denominator

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

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

Clear the rational surd expression to form a linear relation in terms of radicals

2

Expand the terms and isolate

\sqrt{x}

\sqrt{x} = 3\sqrt{3}

Group terms involving

\sqrt{x}

on one side and constant surds on the other side

3

Square both sides of the simplified equation

x = (3\sqrt{3})^2 = 9 \times 3 = 27

Eliminate the radical over

x

to obtain the integer solution

Key Concept

Solving algebraic surd equations using rationalization concepts and properties of radicals

Estimated Time:1m 30s

Surds and Rationalization of Denominators

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