Exponents and Surds: An Overview

Exponents and surds are very important fundamental mathematics concepts studied at various grades to build a strong foundation for advanced topics. In grade 11, students dive deeper into exponents and surds, exploring their properties and applications through detailed notes and examples. There are many resources that a student can find with names like "exponents and surds grade 11 pdf" and "exponents and surds grade 11 notes pdf". These rescources provide comprehensive explanations and practice problems. Grade 12 students will face more complex problems, as seen in "exponents and surds grade 12 questions and answers pdf," which help solidify their understanding. For students in lower grades, such as those in grades 9 and 10, introductory materials like “exponents and surds grade 10 pdf”, “exponents and surds grade 9 pdf” and also exponents and surds worksheets are available to introduce these concepts gradually. Practice problems and solutions, such as those in “exponents and surds questions and solutions”, "indices problems and solutions" and "indices problems and solutions pdf," are essential for mastering the manipulation and simplification of expressions involving exponents and surds.

Exponents of a real number

Exponents of a real number represent the multiplication of the base number. They can be positive, negative, zero or fractional. Some of the types are:

• Positive exponents

• Negative exponents

• Zero exponents

• Rational exponents

We have discussed thoroughly these exponents below.

Positive Exponents

If an exponent indicates how many times a number is multiplied by itself, it is called a Positive exponent.

Example:

3 × 3 × 3 = 33 = 9

Here 33 shows how many times the number 3 is multiplied by itself. So it is a Positive exponent.

Negative Exponents

If an exponent represents the reciprocal of the base raised to the absolute value of the exponent, then it is called a Negative exponent.

Example:

3-3 = $\frac{1}{3^3}=\frac{1}{9}$

Zero Exponent

Any number with an exponent of zero has a value equal to 1.

Example:

50 = 1, 100 = 1, (-3)0 = 0, etc.

Rational Exponents

Rational exponents are also known as Fractional exponents. It represents both roots and powers.

Example: am/n is equivalent to the nth root of a raised to the power m.

163/4 = $(\sqrt[4]{16})^3$ = 23 = 8

Laws of Exponents

Example 1:

Which value among $3^{200},2^{300},$ and $7^{100}$ is the largest?

$3^{200}=(3^{2})^{100}=9^{100}$

$2^{300}=(2^{3})^{100}=8^{100}$

$7^{100}=(7^{1})^{100}=7^{100}$

Thus, the largest number is $3^{200}$.

Example 2:
If $\left (\sqrt{5} \right)^{7}\div \left (\sqrt{5} \right)^{5}=5^{p},$ then the value of $p$ is:

Given: $\frac{(\sqrt{5})^7}{(\sqrt{5})^5} = 5^p$

$⇒(\sqrt{5})^{7 - 5} = 5^p$

$⇒(\sqrt{5})^2 = 5^p$

$⇒5^{\frac{1}{2}×2} = 5^{p}$

$⇒5^1=5^p$

Comparing both sides, we get,

$\therefore p=1$

Laws of Rational Exponents

Rational exponents, also known as fractional exponents, follow similar laws to integer exponents. These laws help simplify and solve expressions involving roots and powers.

Example:

The value of $\frac{(243)^\frac{n}{5}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}$ is:

Given: $\frac{(243)^\frac{n}{5}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}$

= $\frac{(3^5)^\frac{n}{5}\times 3^{2n+1}}{3^{2n}\times 3^{n-1}}$

= $\frac{3^n\times 3^{2n+1}}{3^{2n}\times 3^{n-1}}$

= $\frac{3^{3n+1}}{3^{3n-1}}$

= $\frac{3^{3n}×3^1}{3^{3n}×3^{-1}}$

= ${3^{2}}$

= $9$

Hence, the value is 9.

What is a Surd?

A surd is an irrational number which can not be simplified into a whole number or rational number after the square root or cube root of the number. It is not a perfect square or perfect cube number.

Example: $\sqrt{2}, \sqrt[3]{10}, \sqrt[4]{20},$ etc.

Types of Surds

Simple Surds

Basic irrational numbers that are roots of integers that consist of only one term are called Simple Surds.

Example: $\sqrt2, \sqrt5,\sqrt7,$ etc.

Pure Surds

Surds that are entirely in the root form without any rational number coefficient are called Pure surds. Pure surds can not be written in a fraction form.

Example: $\sqrt{11}, \sqrt{13}, \sqrt{21},$ etc.

Mixed Surds

Surds that are mixed with rational numbers and irrational numbers are called Mixed Surds.

Example: $\sqrt[4]{7}, \sqrt[5]{3}, \sqrt[9]{6},$ etc.

Compound Surds

The sum or difference of two or more surds is called Compound surd.

Example: $(\sqrt3 + \sqrt7 + \sqrt{17}), (\sqrt8 - \sqrt[5]{3}),$ etc

Similar Surds

Surds with the same radicand(number under the roots) are called Similar surds. So it can be easily added or subtracted by combining the exponents.

Example: $2\sqrt5 + 3\sqrt5 = 5\sqrt5$

Binomial Surds

A compound surd consisting of only two terms is called a Binomial Surd.

Example: $(\sqrt6 - \sqrt[5]{9}), (\sqrt5 + \sqrt[2]{12}),$ etc.

Laws of Surds

Simplification Surds

Simplification of surds means converting the surds into a simple form, where the number inside the roots or radicands contains no square factors other than 1.

This process needs some steps which are explained below.

Step 1: Break down the number inside the roots into its prime factors.

Step 2: Pair the prime factors.

Step 3: Take out pairs of prime factors from under the square root, as they become whole numbers.

Example:

Simplify $\sqrt{24}$.

$\sqrt{24}=\sqrt{2×2×2×3}=\sqrt{2^2×2×3}=2\sqrt{2×3}=2\sqrt6$

Simplify $4\sqrt{75}$.

$4\sqrt{75}=4\sqrt{5×5×3}=4\sqrt{5^2×3}=4×5\sqrt{3}=20\sqrt3$

Tips and Tricks

• Any number with an exponent of zero has a value equal to 1.

• am/n is equivalent to the nth root of a raised to the power m.

• $\sqrt a × \sqrt b=\sqrt{ab}$

• Let b be a rational number.

$\frac{1}{\sqrt b}$

Then,

$\frac{1×\sqrt b}{\sqrt b×\sqrt b} = \frac{\sqrt b}{b}$

• Conjugate of $(a+\sqrt b)$ is $(a-\sqrt b)$.

• am × an = amn

• $\frac{a^m}{a^n}$ = am – n

• For surds, simplify the radicand, combine like terms, rationalize denominators, and use surd properties for simplification.

• For Exponents, memorize and apply the laws of exponents, and practice simplifying complex expressions.

Practice Questions

Q1. If $(2^{3})^{2} = 4^{x}$, then $3^{x}$ is equal to:

1. 3

2. 6

3. 9

4. 27

Hint: Use the indices rule: if $a^x = a^y$, then $x=y$

Answer:

Given: $(2^{3})^{2} = 4^{x}$

⇒ $2^{6} = 2^{2x}$

⇒ $6 = 2x$ (If $a^x = a^y$, then $x=y$)

⇒ $x = 3$

$\therefore$ $3^{x} = 3^{3} = 27$

Hence, the correct answer is 27.

Q2. If $4^{(x+y)} = 256$ and $(256)^{(x–y)} = 4$, what are the values of $x$ and $y?$

1. $\frac{17}{8}$ and $\frac{15}{8}$

2. $\frac{17}{4}$ and $\frac{15}{4}$

3. $\frac{9}{17}$ and $\frac{15}{17}$

4. $\frac{8}{17}$ and $\frac{8}{15}$

Hint: Use the indices rule: if $a^x = a^y$, then $x=y$

Answer:

Given: $4^{(x+y)}=256$

⇒ $4^{(x+y)} = 4^4$

⇒ $x+y = 4$

⇒ $x = 4-y$-----(1)

And $(256)^{(x-y)}=4$

⇒ $(4^4)^{(x-y)}=4^1$

⇒ $4x-4y =1$-----(2)

Putting the value of $x$ from equation (1), we get,

$4(4-y)-4y = 1$

⇒ $16-4y-4y =1$

$\therefore y =\frac{15}{8}$

Now, substituting the value of $y$ in equation (1), we get,

$x = 4-\frac{15}{8}$

$\therefore x = \frac{17}{8}$

Hence, the correct answer is $\frac{17}{8}$ and $\frac{15}{8}$.

Q3. The value of $(x^{\frac{1}{3}}+x^{-\frac{1}{3}})(x^{\frac{2}{3}}-1+x^{-\frac{2}{3}})$ is:

1. $x^{-1}+x^{\frac{2}{3}}$

2. $x+x^{-\frac{1}{3}}$

3. $x^{\frac{1}{3}}+x^{–1}$

4. $x+x^{–1}$

Hint: Multiply both the terms and use the rule of indices: $a^x \times a^y = a^{x+y}$

Answer:

Given: $(x^{\frac{1}{3}}+x^{-\frac{1}{3}})(x^{\frac{2}{3}}-1+x^{-\frac{2}{3}})$

$=x^{\frac{1}{3}}×x^{\frac{2}{3}}-x^{\frac{1}{3}}+x^{\frac{1}{3}}×x^{-\frac{2}{3}}+x^{-\frac{1}{3}}×x^{\frac{2}{3}}-x^{-\frac{1}{3}}+x^{-\frac{1}{3}}×x^{-\frac{2}{3}}$

$=x^{\frac{1}{3}+\frac{2}{3}}-x^{\frac{1}{3}}+x^{\frac{1}{3}-\frac{2}{3}}+x^{-\frac{1}{3}+\frac{2}{3}}-x^{-\frac{1}{3}}+x^{-\frac{1}{3}-\frac{2}{3}}$

$=x^1-x^{\frac{1}{3}}+x^{-\frac{1}{3}}+x^{\frac{1}{3}}-x^{-\frac{1}{3}}+x^{-1}$

$=x+x^{-1}$

Hence, the correct answer is $x+x^{-1}$.

Q4. If the numbers $\sqrt[3]{9}, \sqrt[4]{20}$ and $\sqrt[6]{25}$ are arranged in ascending order, then the right arrangement is:

1. $\sqrt[6]{25}<\sqrt[4]{20}< \sqrt[3]{9}$

2. $\sqrt[3]{9}<\sqrt[4]{20}<\sqrt[6]{25}$

3. $\sqrt[4]{20}<\sqrt[6]{25}<\sqrt[3]{9}$

4. $\sqrt[6]{25}<\sqrt[3]{9}<\sqrt[4]{20}$

Hint: Simplify the given values by taking the least common multiple (LCM) of the roots to make roots equal, then compare to find the desired value.

Answer:

Given: The numbers are $\sqrt[3]{9}$, $\sqrt[4]{20}$, $\sqrt[6]{25}$.

LCM of 3, 4, and 6 = 12

$⇒ \sqrt[3]{9}=9^{\frac{1}{3}}=9^{\frac{4}{12}}=\sqrt[12]{6561}$

$⇒ \sqrt[4]{20}=20^{\frac{1}{4}}=20^{\frac{3}{12}}=\sqrt[12]{8000}$

$⇒ \sqrt[6]{25}=25^{\frac{1}{6}}=25^{\frac{2}{12}}=\sqrt[12]{625}$

By comparing these values, we get:

$\sqrt[12]{625}<\sqrt[12]{6561}<\sqrt[12]{8000}$

$⇒\sqrt[6]{25}<\sqrt[3]{9}<\sqrt[4]{20}$

Hence, the correct answer is $\sqrt[6]{25}<\sqrt[3]{9}<\sqrt[4]{20}$.

Q5. What is the value of $\left[\frac{12}{(\sqrt5+\sqrt3)}+\frac{18}{(\sqrt{5}-\sqrt3)}\right]$?

1. $15(\sqrt5–\sqrt3)$

2. $3(5\sqrt5+\sqrt3)$

3. $15(\sqrt5+\sqrt3)$

4. $3(3\sqrt5+\sqrt3)$

Hint: Use the formula: $(\sqrt a+\sqrt b)(\sqrt a–\sqrt b) = a-b$

Answer:

Given:

$\left[\frac{12}{(\sqrt5+\sqrt3)}+\frac{18}{(\sqrt{5}-\sqrt3)}\right]$

= $\frac{12\sqrt5-12\sqrt 3+18\sqrt5+18\sqrt 3}{(\sqrt5+\sqrt3)(\sqrt{5}-\sqrt3)}$

= $\frac{30\sqrt5+6\sqrt3}{5-3}$

= $\frac{6(5\sqrt5+\sqrt3)}{2}$

= $3(5\sqrt5+\sqrt3)$

Hence, the correct answer is $3(5\sqrt5+\sqrt3)$.

Q6. The quotient when $10^{100}$ is divided by $5^{75}$ is:

1. $2^{25}×10^{75}$

2. $10^{25}$

3. $2^{75}$

4. $2^{75}×10^{25}$

Hint: Solve by using the formula: $\frac{a^{m}}{a^{n}}=a^{(m-n)}$ and simplify to find the unknown value.

Answer:

Given: $10^{100}$ is divided by $5^{75}$.

= $\frac{10^{100}}{5^{75}}$

= $\frac{(2×5)^{100}}{5^{75}}$

= $\frac{2^{100}×5^{100}}{5^{75}}$

= $2^{100}×\frac{5^{100}}{5^{75}}$

By using, $\frac{a^{m}}{a^{n}}=a^{(m–n)}$

= $2^{100}×5^{(100-75)}$

= $2^{100}×5^{25}$

= $2^{75}×2^{25}×5^{25}$

= $2^{75}×10^{25}$

Hence, the correct answer is $2^{75}×10^{25}$.

Q7. What is the value of the given expression?

$\frac{4^{a + 4}- 5 \times 4^{a + 2}}{15 \times 4^a - 2^2 \times 4^a}$

1. 16

2. 64

3. 20

4. 24

Hint: Use the information below to solve the problem.

$x^{ab}=x^a×x^b$

Answer:

$\frac{4^{a+ 2}(4^{2} - 5 \times 1)}{4^{a}(15 - 4)}$

$=\frac{4^{a + 2}(11)}{4^{a}(11)}$

$=\frac{4^{a}\times4^{2}(11)}{4^{a}(11)}$

$=4^2$

$=16$

Hence, the correct answer is 16.

Q8. If $a=b^p,b=c^q,c=a^r$, then $pqr$ is:

1. 1

2. 0

3. -1

4. abc

Hint: Use the values of $b$ and $c$ in the equation $a=b^p$.

Answer:

Given:

$a=b^p,b=c^q,c=a^r$

$a=b^p$

⇒ $a=(c^q)^p$

⇒ $a=(a^r)^{pq}$

⇒ $a^1=(a)^{pqr}$

$\therefore pqr=1$

Hence, the correct answer is 1.

Q9. When xm is multiplied by xn product is 1. The relation between m and n is:

1. mn = 1

2. m = n

3. m + n = 1

4. m = – n

Hint: Use the following formula to solve the question:

xm × xn = xm + n

Answer:

xm × xn = 1

⇒ xm + n = x0

⇒ m + n = 0

⇒ m = –n

Hence, the correct answer is m = – n.

Q10. If $2^{2x-y}=16$ and $2^{x+y}=32$, the value of $xy$ is:

1. 2

2. 4

3. 6

4. 8

Hint: If am = an, then m = n.

Answer:

Given: $2^{2x-y}=16$

$⇒2^{2x-y}=2^4$

$⇒2x-y=4$--------------------(1)

Again, $2^{x+y}=32$

$⇒2^{x+y}=2^5$

$⇒x+y=5$ ---------------------(2)

Solving equations (1) and (2), we get:

$x=3$ and $y= 2$

$\therefore xy=3×2=6$

Hence, the correct answer is $6$.