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SURDS

A surd is a square root which cannot be reduced to a rational number.

For example, is not a surd.

However is a surd.

If you use a calculator, you will see that and we will need to round the answer correct to a few decimal places. This makes it less accurate.

If it is left as , then the answer has not been rounded, which keeps it exact.

Here are some general rules when simplifying expressions involving surds.

a^m x aⁿ = a^m⁺ⁿ

a^m	= a^m^–ⁿ
aⁿ	= a^m^–ⁿ

(a^m)ⁿ = a^mn

(ab)ⁿ = aⁿbⁿ

a	n	=	aⁿ
b	n	=	bⁿ

a⁰ = 1

Questions

Level-I

(17)^3.5 x (17)^? = 17⁸

A.	2.29
B.	2.75
C.	4.25
D.	4.5

If		a		x – 1	=		b		x – 3	, then the value of x is:
		b					a

Given that 10^0.48 = x, 10^0.70 = y and x^z = y², then the value of z is close to:

A.	1.45
B.	1.88
C.	2.9
D.	3.7

If 5^a = 3125, then the value of 5⁽^a^{– 3)} is:

A.	25
B.	125
C.	625
D.	1625

If 3⁽^x^–^y⁾ = 27 and 3⁽^x⁺^y⁾ = 243, then x is equal to:

A.	0
B.	2
C.	4
D.	6

.6.

(256)^0.16 x (256)^0.09 = ?

A.	4
B.	16
C.	64
D.	256.25

The value of [(10)¹⁵⁰ ÷ (10)¹⁴⁶]

A.	1000
B.	10000
C.	100000
D.	10⁶

1	+	1	+	1	= ?
1 + x⁽^b^–^a⁾ + x⁽^c^–^a⁾		1 + x⁽^a^–^b⁾ + x⁽^c^–^b⁾		1 + x⁽^b^–^c⁾ + x⁽^a^–^c⁾

A.	0
B.	1
C.	x^a^–^b^–^c
D.	None of these

(25)^7.5 x (5)^2.5 ÷ (125)^1.5 = 5^?

A.	8.5
B.	13
C.	16
D.	17.5
E.	None of these

10.

(0.04)^-1.5 = ?

A.	25
B.	125
C.	250
D.	625

Level-II

11.

(243)ⁿ^/5 x 3²ⁿ^{+ 1}	= ?
9ⁿ x 3ⁿ^{– 1}

A.	1
B.	2
C.	9
D.	3ⁿ

12.

1	+	1	= ?
1 + a⁽ⁿ^–^m⁾		1 + a⁽^m^–ⁿ⁾

a^m⁺ⁿ

13.

If m and n are whole numbers such that mⁿ = 121, the value of (m – 1)ⁿ^{+ 1} is:

A.	1
B.	10
C.	121
D.	1000

14.

x^b

(b + c – a)

x^c

(c + a – b)

x^a

⁽^a⁺^b^–^c⁾

= ?

x^c

x^a

x^b

A.	x^abc
B.	1
C.	x^ab⁺^bc⁺^ca
D.	x^a⁺^b⁺^c

15. If 5√5 * 5³ ÷ 5^-3/2= 5^a+2 , the value of a is:
A. 4
B. 5
C. 6
D. 8

16.(132)⁷ ×(132)^? =(132)^11.5.

A. 3
B. 3.5
C. 4
D. 4.5

17. (ab)x−2=(ba)x−7. What is the value of x ?

A. 3
B. 4
C. 3.5
D. 4.5

18. (0.04)^-2.5 = ?

A. 125
B. 25
C. 3125
D. 625

Answers

Level-I

Answer:1 Option D

Explanation:

Let (17)^3.5 x (17)^x = 17⁸.

Then, (17)^{3.5 +}^x = 17⁸.

3.5 + x = 8

x = (8 – 3.5)

x = 4.5

Answer:2 Option C

Explanation:

Given	a		x – 1	=		b		x – 3
	b					a

x – 1

-(x – 3)

(3 – x)

x – 1 = 3 – x

2x = 4

x = 2.

Answer:3 Option C

Explanation:

x^z = y² 10^(0.48^z⁾ = 10^{(2 x 0.70)} = 10^1.40

0.48z = 1.40

z =	140	=	35	= 2.9 (approx.)
	48		12

Answer:4 Option A

Explanation:

5^a = 3125 5^a = 5⁵

a = 5.

5⁽^a^{– 3)} = 5^{(5 – 3)} = 5² = 25.

Answer:5 Option C

Explanation:

3^x^–^y = 27 = 3³ x – y = 3 ….(i)

3^x⁺^y = 243 = 3⁵ x + y = 5 ….(ii)

On solving (i) and (ii), we get x = 4

Answer:6 Option A

Explanation:

(256)^0.16 x (256)^0.09 = (256)^{(0.16 + 0.09)}

= (256)^0.25

= (256)^(25/100)

= (256)^(1/4)

= (4⁴)^(1/4)

= 4^4(1/4)

= 4¹

= 4

Answer:7 Option B

Explanation:

(10)¹⁵⁰ ÷ (10)¹⁴⁶ =	10¹⁵⁰
	10¹⁴⁶

= 10^{150 – 146}

= 10⁴

= 10000.

Answer:8 Option B

Explanation:

Given Exp. =

	1 +	x^b	+	x^c
		x^a		x^a

	1 +	x^a	+	x^c
		x^b		x^b

	1 +	x^b	+	x^a
		x^c		x^c

=	x^a	+	x^b	+	x^c
	(x^a + x^b + x^c)		(x^a + x^b + x^c)		(x^a + x^b + x^c)

=	(x^a + x^b + x^c)
	(x^a + x^b + x^c)

= 1.

Answer:9 Option B

Explanation:

Let (25)^7.5 x (5)^2.5 ÷ (125)^1.5 = 5^x.

Then,	(5²)^7.5 x (5)^2.5	= 5^x
	(5³)^1.5

	5^{(2 x 7.5)} x 5^2.5	= 5x
	5^{(3 x 1.5)}

	5¹⁵ x 5^2.5	= 5^x
	5^4.5

5^x = 5^{(15 + 2.5 – 4.5)}

5^x = 5¹³

x = 13.

Answer:10 Option B

Explanation:

(0.04)^-1.5 =		4		-1.5
		100

=		1		-(3/2)
		25

= (25)^(3/2)

= (5²)^(3/2)

= (5)^{2 x (3/2)}

= 5³

= 125.

Level-II

Answer:11 Option C

Explanation:

Given Expression

=	(243)⁽ⁿ^/5) x 3²ⁿ^{+ 1}
	9ⁿ x 3ⁿ^{– 1}

=	(3⁵)⁽ⁿ^/5) x 3²ⁿ^{+ 1}
	(3²)ⁿ x 3ⁿ^{– 1}

=	(3^{5 x (}ⁿ^/5) x 3²ⁿ^{+ 1})
	(3²ⁿ x 3ⁿ^{– 1})

=	3ⁿ x 3²ⁿ^{+ 1}
	3²ⁿ x 3ⁿ^{– 1}

=	3⁽ⁿ^{+ 2}ⁿ^{+ 1)}
	3⁽²ⁿ⁺ⁿ^{– 1})

=	3³ⁿ^{+ 1}
	3³ⁿ^{– 1}

= 3⁽³ⁿ^{+ 1 – 3}ⁿ^{+ 1)} = 3² = 9.

Answer:12 Option C

Explanation:

	1 +	aⁿ
		a^m

	1 +	a^m
		aⁿ

1 + a⁽ⁿ^–^m⁾

1 + a⁽^m^–ⁿ⁾

=	a^m	+	aⁿ
	(a^m + aⁿ)		(a^m + aⁿ)

=	(a^m + aⁿ)
	(a^m + aⁿ)

= 1.

Answer:13 Option D

Explanation:

We know that 11² = 121.

Putting m = 11 and n = 2, we get:

(m – 1)ⁿ^{+ 1} = (11 – 1)^{(2 + 1)} = 10³ = 1000.

Answer:14 Option B

Explanation:

Given Exp.

= x⁽^b^–^c⁾⁽^b⁺^c^–^a⁾ . x⁽^c^–^a⁾⁽^c⁺^a^–^b⁾ . x⁽^a^–^b⁾⁽^a⁺^b^–^c⁾

= x⁽^b^–^c⁾⁽^b⁺^c^{) –}^a⁽^b^–^c⁾ . x⁽^c^–^a⁾⁽^c⁺^a^{) –}^b⁽^c^–^a⁾
. x⁽^a^–^b⁾⁽^a⁺^b^{) –}^c⁽^a^–^b⁾

= x⁽^b^2 –^c^2 +^c^2 –^a^2 +^a^2 –^b²⁾ . x^–^a⁽^b^–^c^{) –}^b⁽^c^–^a^{) –}^c⁽^a^–^b⁾

= (x⁰ x x⁰)

= (1 x 1) = 1.

Answer:15 option C

Answer:16

Explanation

am.an=am+n

(132)⁷ × (132)^x = (132)^11.5

=> 7 + x = 11.5

=> x = 11.5 – 7 = 4.5

Answer:17

Explanation:

an=1a−n

(ab)x−2=(ba)x−7⇒(ab)x−2=(ab)−(x−7)⇒x−2=−(x−7)⇒x−2=−x+7⇒x−2=−x+7⇒2x=9⇒x=92=4.5

Answer:18

Explanation:

a−n=1/an

(0.04)−2.5=(1/.04)2.5=(100/4)2.5=(25)2.5=(52)2.5=(52)(5/2)=55=3125

Introduction to Surds

A surd is a number that cannot be expressed as a fraction of two integers. Surds are often written as roots, such as $\sqrt{2}$ or $\pi$.

Surds can be simplified by combining them into a single surd. For example, $\sqrt{2} + \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6}$.

Surds can also be added, subtracted, multiplied, and divided. However, it is important to remember that the order of operations must be followed when performing these operations on surds.

For example, to add two surds, the surds must be in the same form. For example, $\sqrt{2} + \sqrt{3}$ can be added, but $\sqrt{2} + 3$ cannot be added.

To subtract two surds, the surds must be in the same form. For example, $\sqrt{2} – \sqrt{3}$ can be subtracted, but $\sqrt{2} – 3$ cannot be subtracted.

To multiply two surds, the product of the two surds is the square root of the product of the two numbers under the radical. For example, $\sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6}$.

To divide two surds, the quotient of the two surds is the square root of the quotient of the two numbers under the radical. For example, $\frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}}$.

Surds can also be raised to powers. To raise a surd to a power, the power is applied to both the number under the radical and the radical itself. For example, $(\sqrt{2})^3 = \sqrt{2 \cdot 2 \cdot 2} = 2\sqrt{2}$.

Surds can also be taken to roots. To take the root of a surd, the root is applied to both the number under the radical and the radical itself. For example, $\sqrt[3]{\sqrt{2}} = \sqrt[3]{2}$.

Equations Involving Surds

Equations involving surds can be solved by isolating the surd on one side of the equation and then simplifying the equation. For example, to solve the equation $\sqrt{x} + 2 = 3$, we can subtract 2 from both sides of the equation to get $\sqrt{x} = 1$. Then, we can take the square root of both sides of the equation to get $x = 1$.

Indices

An index is a number that indicates how many times a base number is multiplied by itself. For example, $2^3 = 2 \cdot 2 \cdot 2 = 8$, and $3^2 = 3 \cdot 3 = 9$.

The base number is the number that is being multiplied by itself. The exponent is the number that indicates how many times the base number is multiplied by itself.

Indices can be added, subtracted, multiplied, and divided. However, it is important to remember that the order of operations must be followed when performing these operations on indices.

For example, to add two indices, the bases of the indices must be the same. For example, $2^3 + 3^2 = 8 + 9 = 17$.

To subtract two indices, the bases of the indices must be the same. For example, $2^3 – 3^2 = 8 – 9 = -1$.

To multiply two indices, the bases of the indices must be the same. For example, $2^3 \cdot 3^2 = 8 \cdot 9 = 72$.

To divide two indices, the bases of the indices must be the same. For example, $\frac{2^3}{3^2} = \frac{8}{9} = \frac{8}{9}$.

Laws of Indices

There are several laws of indices that can be used to simplify expressions involving indices. These laws are as follows:

The product of two powers with the same base is the power of the product of the bases, with the same exponent. For example, $a^m \cdot a^n = a^{m+n}$.
The quotient of two powers with the same base is the power of the quotient of the bases, with the same exponent. For example, $\frac{a^

Here are some frequently asked questions and short answers about the topic of Surds and Indices.

What is a surd?
A surd is an irrational number that cannot be expressed as a fraction of two integers. Examples of surds include $\sqrt{2}$, $\pi$, and $e$.
What is an index?
An index is a number that indicates how many times a base number is multiplied by itself. For example, $2^3$ is equal to $2\times2\times2$, or 8.
How do you add, subtract, multiply, and divide surds?
To add or subtract surds, they must have the same index. To multiply surds, you multiply the radicands and the indices. To divide surds, you flip the second surd and multiply.
What are some properties of surds?
Some properties of surds include:
- The square of a surd is always a surd.
- The cube of a surd is always a surd.
- The product of two surds is a surd.
- The quotient of two surds is a surd.
What are some applications of surds?
Surds have many applications in mathematics, physics, and engineering. For example, surds can be used to solve quadratic equations, to find the roots of polynomials, and to calculate the areas and volumes of geometric shapes.
What are some common mistakes people make when working with surds?
Some common mistakes people make when working with surds include:
- Forgetting to simplify surds before performing arithmetic operations.
- Not realizing that the square of a surd is always a surd.
- Not realizing that the cube of a surd is always a surd.
- Not realizing that the product of two surds is a surd.
- Not realizing that the quotient of two surds is a surd.
Where can I learn more about surds?
You can learn more about surds in a variety of places, including:
- Your math textbook.
- A math tutor.
- A math website.
- A math forum.
- A math library.

Sure. Here are some MCQs without mentioning the topic Surds And Indices:

What is the value of $x^2 + 2x – 8$ when $x = 3$?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
What is the value of $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^{10}}$?
(A) $\frac{1}{2}$
(B) $\frac{1}{4}$
(C) $\frac{1}{8}$
(D) $\frac{1}{16}$
(E) $\frac{1}{32}$
What is the value of $\sqrt{25}$?
(A) 5
(B) 7
(C) 10
(D) 14
(E) 17
What is the value of $\sqrt[3]{27}$?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 18
What is the value of $\log_{10} 100$?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
What is the value of $\log_{10} 0.01$?
(A) -2
(B) -3
(C) -4
(D) -5
(E) -6
What is the value of $e^2$?
(A) 7.38905609893065
(B) 27.18281828459045
(C) 72.7253908203125
(D) 125.9765625
(E) 218.75
What is the value of $\sin 30^\circ$?
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\frac{\sqrt{2}}{2}$
(D) $\frac{1}{4}$
(E) $\frac{1}{8}$
What is the value of $\cos 60^\circ$?
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\frac{\sqrt{2}}{2}$
(D) $\frac{1}{4}$
(E) $\frac{1}{8}$
What is the value of $\tan 45^\circ$?
(A) $\frac{1}{\sqrt{2}}$
(B) $\sqrt{2}$
(C) $\frac{\sqrt{3}}{2}$
(D) $\frac{1}{2}$
(E) 1

I hope these questions are helpful!