Exponents and Surds: Overview, Principles, Examples

Exponents and Surds: When a number is multiplied by itself several times, it can be written in a short form with the help of exponents. Example: \(4 \times 4 \times 4=4^{3}\). Here, \(4^{3}\) is called an exponential expression. The number multiplied by itself, again and again, is the base. The number of times the number appears or is multiplied is the exponent. In \({4^3},4\) is the base and \(3\) is the exponent.

The values in the square root or cube root or any other roots, which cannot be further simplified into whole numbers or integers, are known as a surd. Example: The value of \(\sqrt 2 = 1.4142135 \ldots .\) In this article, we will study the definition of exponents, surds and several laws of exponents and solve some example problems on surds and exponents. Scroll down to learn more!

Exponents

The continued product of a number multiplied by itself several times can be written as the number raised to the power of a natural number, equal to the number of times the number is multiplied with itself.

For example, \(2×2×2\) can be written as \(2^{3}\) and it is read as \(2\) raised to the power \(3\) or third power of \(2\). In \(2^{3}\), we call \(2\) as the base and \(3\) as the exponent.

Exponents of a Real Number

The exponents of a real number are as follows:

Positive Exponent: For any real number \(a\) and a positive integer \(n\), we define \({a^n}\) as

\(a^{n}=a \times a \times a \times a \times \ldots \ldots \times a( n\, {\rm{factors}} )\)
\({a^n}\) is called the \({{\rm{n}}^{{\rm{th}}}}\) power of \(a\).

The real number \(a\) is called the base and \(n\) is called the exponent of the \({{\rm{n}}^{{\rm{th}}}}\) power of \(a.\)
Example: \(3^{3}=3 \times 3 \times 3=27\)
\(\left(\frac{-2}{4}\right)^{4}=\frac{-2}{4} \times \frac{-2}{4} \times \frac{-2}{4} \times \frac{-2}{4}=\frac{16}{256}\)

Negative Exponent: For any non-zero real number \(a\) and a positive integer \(n\), we define \(a^{-n}=\frac{1}{a^{n}}\)
Example: \(\left(\frac{1}{5}\right)^{-2}=\frac{1}{(1 / 5)^{2}}=\frac{1}{1 / 5 \times 1 / 5}=\frac{1}{1 / 25}=25\)
\((2)^{-3}=\frac{1}{2^{3}}=\frac{1}{2 \times 2 \times 2}=\frac{1}{8}\)

Laws of Exponents

In this section, we will learn about various laws of exponents.

First Law

If \(a\) is any non-zero rational number and \(m, n\) are natural numbers, then \(a^{m} \times a^{n}=a^{(m+n)}\)

Also, if \(a\) is any non-zero rational number and \(m, n, p\) are natural numbers, then \(a^{m} \times a^{n} \times a^{p}=a^{(m+n+p)}\)

Example: \(4^{2} \times 4^{4}=(4 \times 4) \times(4 \times 4 \times 4 \times 4)=4 \times 4 \times 4 \times 4 \times 4 \times 4=4^{6}=4^{(2+4)}\)

Second Law

If \(a\) is any non-zero rational number and \(m, n\) are natural numbers such that \(m>n\), then \(a^{m} \div a^{n}=a^{(m-n)}\) or \(\frac{a^{m}}{a^{n}}=a^{(m-n)}\)

Example: \(\frac{3^{7}}{3^{4}}=\frac{3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3}{3 \times 3 \times 3 \times 3}=3 \times 3 \times 3=3^{3}=3^{(7-4)}\)

Third Law

If \(a\) is any non-zero rational number and \(m, n\) are natural numbers, then

\(\left(a^{m}\right)^{n}=a^{(m \times n)}=\left(a^{n}\right)^{m}\)

Example: \(\left(2^{2}\right)^{3}=2^{2} \times 2^{2} \times 2^{2}=2^{(2+2+2)}=2^{6}=2^{(2 \times 3)}\)

Fourth Law

If \(a, b\) are non-zero rational numbers and \(n\) is a natural number, then \(a^{n} \times b^{n}=(a b)^{n}\)

Example: \({2^4} \times {5^4} = (2 \times 2 \times 2 \times 2) \times (5 \times 5 \times 5 \times 5) = (2 \times 5) \times (2 \times 5) \times (2 \times 5) \times (2 \times 5)\)
\(= 10 \times 10 \times 10 \times 10 = {10^4} = {(2 \times 5)^4}\)

Fifth Law

If \(a\) and \(b\) are non-zero rational numbers and \(n\) is a natural number, then \(\frac{a^{n}}{b^{n}}=\left(\frac{a}{b}\right)^{n}\)

Example: \(\frac{6^{3}}{5^{3}}=\frac{6 \times 6 \times 6}{5 \times 5 \times 5}=\frac{6}{5} \times \frac{6}{5} \times \frac{6}{5}=\left(\frac{6}{5}\right)^{3}\)

Power with Exponent Zero

Let us now find the value of a power of a non-zero rational number when its exponent is zero.

We know that \(\frac{4^{4}}{4^{4}}=\frac{4 \times 4 \times 4 \times 4}{4 \times 4 \times 4 \times 4}=1 \ldots (i)\)

Also, \(\frac{{{4^4}}}{{{4^4}}} = {4^{(4 – 4)}} = {4^0} \ldots \ldots (ii)\)

From (i) and (ii), we can write as \({4^0} = 1\)
Therefore, for any non-zero rational number \(a\) we have \(a^{0}=1\)

Rational Exponents of a Real Number

In the above section, we have studied the integral power of a real number. In this section, we will define rational exponents of a real number.

Principal nth Root of a Positive Real Number

If \(a\) is a positive real number and \(n\) is a positive integer, then the principal \({{\rm{n}}^{{\rm{th}}}}\) root of \(a\) is the unique positive real number \(x\) such that \({x^n} = a.\)

The principal \({{\rm{n}}^{{\rm{th}}}}\) root of a positive real number \(a\) is denoted by \(a^{1 / n}\) or, \(\sqrt[n]{a}\)

For example \((16)^{1 / 2}=4\), because \(4^{2}=16\)

Principal nth Root of a Negative Real Number

If \(a\) is a negative real number and \(n\) is an odd positive integer, then the principal \({{\rm{n}}^{{\rm{th}}}}\) root of \(a\) is defined as \(-|a|^{1 / n}\) i.e., the principal \({{\rm{n}}^{{\rm{th}}}}\) root of \(a\) is minus of the principal \({{\rm{n}}^{{\rm{th}}}}\) root of \(|a|\)

For example: \((-8)^{1 / 3}=-|-8|^{1 / 3}=-8^{1 / 3}=-2\)

If \(a\) is a negative real number and \(n\) is an even positive integer, then the principal \({{\rm{n}}^{{\rm{th}}}}\) root of \(a\) is not defined because an even power of a real number is always positive. Therefore, \((-9)^{1 / 2}\) is a meaningless quantity if we confine ourselves to the set of real numbers only. But, \((-9)^{1 / 2}\) is meaningful in the set of complex numbers which we will study in higher classes.

Rational Exponents

For any positive real number \(a\) and a rational number \(\frac{p}{q},\) where \(q>0\), we define \(a^{p / q}=\left(a^{p}\right)^{1 / q}\) That is, \(a^{p / q}\) is the principal \({{\rm{q}}^{{\rm{th}}}}\) root of \(a^{p}\).

For example \(4^{3 / 2}=\left(4^{3}\right)^{1 / 2}=(64)^{1 / 2}=8\)

Laws of Rational Exponents

Using the laws of exponents and the definitions of \(a^{1 / q}\) and \(a^{p / q}, q>0\), it can be shown that the following laws hold the rational exponents.

1. \(a^{m} \times a^{n}=a^{(m+n)}\)
2. \(a^{m} \div a^{n}=a^{(m-n)}\)
3. \(\left(a^{m}\right)^{n}=a^{(m \times n)}=\left(a^{n}\right)^{m}\)
4. \(a^{-n}=\frac{1}{a^{n}}\)
5. \(a^{m / n}=\left(a^{m}\right)^{1 / n}=\left(a^{n}\right)^{1 / m}\) i.e., \(\sqrt[n]{a^{m}}=(\sqrt[n]{a})^{m}\)
6. \(a^{n} \times b^{n}=(a b)^{n}\)
7. \(\frac{a^{n}}{b^{n}}=\left(\frac{a}{b}\right)^{n}\)

Surds

The square roots(or cube roots or so) of numbers that cannot be simplified into a whole number or rational number are called surds. It is used to refer to a number that does not have a root.

Example: \(\sqrt{16}, \sqrt[3]{27}\) have roots as answers.
But, \(\sqrt{5}, \sqrt{19}, \sqrt[3]{3}\) does not have proper roots.

Also, the surds shown in the above example can be expressed in exponential with fractions as powers.
\(\sqrt{5}=5^{1 / 2}=\sqrt[3]{3}=3^{1 / 3}\)

There are three types of surds. They are,

1. Pure surds
2. Mixed surds
3. Compound surds

1. Pure Surds: The surd which has only a single irrational number is called a pure surd.

Example: \(\sqrt{6}, \sqrt[3]{4}, \sqrt[4]{5}\)

2. Mixed Surds: The surd with a mix of a rational number and an irrational number is called a mixed surd.

Example: \(y \sqrt{x}, 5 \sqrt{4}, 9 \sqrt{6}\)

3. Compound Surds: The surd, made up of two surds, is called a compound surd.

Example: \(\sqrt{x}+\sqrt{y}, \sqrt{3}+\sqrt{2}, \sqrt{8}+2 \sqrt{6}\)

Laws of Surds

Below are the laws for surds.

Law 1: Surds cannot be added
\(\Rightarrow \sqrt{x}+\sqrt{y} \neq \sqrt{x+y}\)

Law 2: Surds cannot be subtracted.
\(\Rightarrow \sqrt{x}-\sqrt{y} \neq \sqrt{x-y}\)

Law 3: Surds can be multiplied
\(\Rightarrow \sqrt{x} \times \sqrt{y}=\sqrt{x \times y}\)

Law 4: Surds can be divided
\(\Rightarrow \frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{\bar{x}}{y}}\)

Law 5: Surds can be expressed in exponential form.
\(\Rightarrow \sqrt{x}=x^{1 / 2}\)

Law 6: Surds can be rationalised.
\(\Rightarrow \frac{x}{\sqrt{y}}=\frac{x}{\sqrt{y}} \times \frac{\sqrt{y}}{\sqrt{y}}=x \frac{\sqrt{y}}{y}\)

Simplification of Surds

Below are the steps to simplify the given surds:

Step 1: Find the prime factors of the number within the root.

Step 2: If it is a square root, write the prime factor outside the root, repeated twice. If it is a cube root, write the prime factor outside the root, repeated thrice.

Step 3: Now, perform the mathematical operation, whichever is required using the like surd.

Example: Add \(\sqrt{12}+\sqrt{48}\)

Solution: \(\sqrt{12}+\sqrt{48}=\sqrt{2 \times 2 \times 3}+\sqrt{4 \times 4 \times 3}=2 \sqrt{3}+4 \sqrt{3}=6 \sqrt{3}\)

Solved Examples on Exponents and Surds

Q.1. Write the expansion of \((2+4 \sqrt{3})^{2}\)
Ans: \((2+4 \sqrt{3})^{2}\)
We know that, \((a+b)^{2}=a^{2}+b^{2}+2 a b\)
So, \((2+4 \sqrt{3})^{2}=2^{2}+(4 \sqrt{3})^{2}+2 \times 2 \times 4 \sqrt{3}\)
\(=4+(16 \times 3)+16 \sqrt{3}\)
\(=4+48+16 \sqrt{3}\)
\(=52+16 \sqrt{3}\)
Therefore, \((2+4 \sqrt{3})^{2}=52+16 \sqrt{3}\)

Q.2. Find the sum of \((2 \sqrt{5}-4 \sqrt{2})\) and \((3 \sqrt{5}+5 \sqrt{2})\)
Ans: We have,
\((2 \sqrt{5}-4 \sqrt{2})+(3 \sqrt{5}+5 \sqrt{2})\)
\(=2 \sqrt{5}-4 \sqrt{2}+3 \sqrt{5}+5 \sqrt{2}\)
\(=2 \sqrt{5}+3 \sqrt{5}+5 \sqrt{2}-4 \sqrt{2}\)
\(=5 \sqrt{5}+\sqrt{2}\)
So, \((2 \sqrt{5}-4 \sqrt{2})+(3 \sqrt{5}+5 \sqrt{2})=5 \sqrt{5}+\sqrt{2}\)

Q.3.  Evaluate the following removing radical signs and negative indices wherever they occur.
(i) \((125)^{-1 / 3}\)
(ii) \(\left(\frac{64}{25}\right)^{-3 / 2}\)
Ans:
(i) We have, \((125)^{-1 / 3}\)
\((125)^{-1 / 3}=\frac{1}{(125)^{1 / 3}}=\frac{1}{\left(5^{3}\right)^{1 / 3}}=\frac{1}{5^{3 \times \frac{1}{3}}}=\frac{1}{5}\)
Therefore, \((125)^{-1 / 3}=\frac{1}{5}\)
(ii) We have, \(\left(\frac{64}{25}\right)^{-3 / 2}\)
\(\left(\frac{64}{25}\right)^{-3 / 2}=\frac{1}{\left(\frac{64}{25}\right)^{3 / 2}}=\left(\frac{25}{64}\right)^{3 / 2}=\left[\left(\frac{5}{8}\right)^{2}\right]^{3 / 2}=\left(\frac{5}{8}\right)^{3}=\frac{125}{512}\)
Therefore, \(\left(\frac{64}{25}\right)^{-3 / 2}=\frac{125}{512}\)

Q.4. Assuming that x, y are positive real numbers, simplify the following:
(i) \(\sqrt{x^{-2} y^{3}}\)
(ii) \((\sqrt{x})^{-2 / 3} \sqrt{y^{4}} \div \sqrt{x y^{-1 / 2}}\)
Ans:
(i) We have, \(\sqrt{x^{-2} y^{3}}\)
\(\sqrt{x^{-2} y^{3}}=\sqrt{\frac{y^{3}}{x^{2}}}\)
\(=\left(\frac{y^{3}}{x^{2}}\right)^{1 / 2}\)
\(=\frac{\left(y^{3}\right)^{1 / 2}}{\left(x^{2}\right)^{1 / 2}}\)
\(=\frac{y^{3 / 2}}{x^{2 / 2}}\)
\(=\frac{y^{3 / 2}}{x}\)
Therefore \(\sqrt{x^{-2} y^{3}}=\frac{y^{3 / 2}}{x}\)

(ii) We have, \((\sqrt{x})^{-2 / 3} \sqrt{y^{4}} \div \sqrt{x y^{-1 / 2}}\)
\((\sqrt{x})^{-2 / 3} \sqrt{y^{4}} \div \sqrt{x y^{-1 / 2}}=\frac{\left(x^{1 / 2}\right)^{-2 / 3}\left(y^{4}\right)^{1 / 2}}{\left(x y^{-1 / 2}\right)^{1 / 2}}\)
\(=\frac{x^{1 / 2 \times-2 / 3} y^{4 \times 1 / 2}}{x^{1 / 2}\left(y^{-1 / 2}\right)^{1 / 2}}\)
\(=\frac{x^{-1 / 3} y^{2}}{x^{1 / 2} y^{-1 / 2 \times 1 / 2}}\)
\(=\frac{x^{-1 / 3} y^{2}}{x^{1 / 2} y^{-1 / 4}}\)
\(=\frac{1}{x^{1 / 3} \times x^{1 / 2}} \times y^{2} \times y^{1 / 4}\)
\(=\frac{1}{x^{(1 / 3)+(1 / 2)}} \times y^{2+(1 / 4)}\)
\(=\frac{1}{x^{\frac{2+3}{6}}} \times y^{\frac{8+1}{4}}\)
\(=\frac{1}{x^{\frac{5}{6}}} \times y^{\frac{9}{4}}\)
\(=\frac{y^{\frac{2}{4}}}{x^{\frac{5}{6}}}\)
Therefore, \((\sqrt{x})^{-2 / 3} \sqrt{y^{4}} \div \sqrt{x y^{-1 / 2}}=\frac{y^{\frac{2}{4}}}{x^{\frac{1}{6}}}\)

Q.5. Find the value of x, if \(5^{x-3} \times 3^{2 x-8}=225\)
Ans: We have,
\(5^{x-3} \times 3^{2 x-8}=225\)
\(\Rightarrow 5^{x-3} \times 3^{2 x-8}=25 \times 9\)
\(\Rightarrow 5^{x-3} \times 3^{2 x-8}=5^{2} \times 3^{2}\)
\(\Rightarrow x-3=2\) and \(2 x-8=2\)
\(\Rightarrow x=5\)

Summary

The square roots of numbers that cannot be simplified into a whole number or rational number are referred to as surds. Furthermore, the three types of surds are pure, mixed, and compound surds. The exponent of a number indicates how many times a particular number is multiplying a number by itself. If \(a\) is a negative real number and \(n\) is an odd positive integer, then the principal \({{\rm{n}}^{{\rm{th}}}}\) root of \(a\) is defined as \(-|a|^{1 / n}\) i.e., the principal \({{\rm{n}}^{{\rm{th}}}}\) root of \(a\) is minus of the principal \({{\rm{n}}^{{\rm{th}}}}\) root of \(|a|\)

FAQs on Exponents and Surds

Q.1. How do you simplify surds with exponents?
Ans: We will first convert the given surd into the exponential form, and then we will use the laws of exponents to solve the given surd.
Example: \((\sqrt{5})^{-3} \times(\sqrt{2})^{-3}=\left(5^{1 / 2}\right)^{-3} \times\left(2^{1 / 2}\right)^{-3}=5^{-3 / 2} \times 2^{-3 / 2}=(5 \times 2)^{-3 / 2}\)
\(=(10)^{-3 / 2}=\frac{1}{(10)^{3 / 2}}=\left(\frac{1}{10^{3}}\right)^{1 / 2}=\frac{1}{(1000)^{1 / 2}}=\left(\frac{1}{1000}\right)^{1 / 2}\)
\(=\left(\frac{10}{10000}\right)^{1 / 2}=\frac{(10)^{1 / 2}}{\sqrt{10000}}=\frac{(10)^{1 / 2}}{100}\)

Q.2. What are the rules of surds?
Ans: Below is the rules for surds.
Rule 1: Surds cannot be added
\(\Rightarrow \sqrt{x}+\sqrt{y} \neq \sqrt{x+y}\)
Rule 2: Surds cannot be subtracted.
\(\Rightarrow \sqrt{x}-\sqrt{y} \neq \sqrt{x-y}\)
Rule 3: Surds can be multiplied
\(\Rightarrow \sqrt{x} \times \sqrt{y}=\sqrt{x \times y}\)
Rule 4: Surds can be divided
\(\Rightarrow \frac{\sqrt{x}}{\bar{y}}=\sqrt{\frac{x}{y}}\)
Rule 5: Surds can be expressed in exponential form.
\(\Rightarrow \sqrt{x}=x^{1 / 2}\)
Rule 6: Surds can be rationalised.
\(\Rightarrow \frac{x}{\sqrt{y}}=\frac{x}{\sqrt{y}} \times \frac{\sqrt{y}}{\sqrt{y}}=x \frac{\sqrt{y}}{y}\)

Q.3. What are examples of surds?
Ans: The square roots(or cube roots or so) of numbers that cannot be simplified into a whole number or rational number are called surds. It is used to refer to a number that does not have a root.
Example: \(\sqrt{5}, \sqrt{19}, \sqrt[3]{3}\) does not have proper roots, which are called surds.

Q.4. What are the five rules of exponents?
Ans: The five rules of exponents are:
Rule 1: If \(a\) is any non-zero rational number and \(m, n\) are natural numbers, then \(a^{m} \times a^{n}=a^{(m+n)}\)
Example: \(4^{2} \times 4^{4}=(4 \times 4) \times(4 \times 4 \times 4 \times 4)=4 \times 4 \times 4 \times 4 \times 4 \times 4=4^{6}=4^{(2+4)}\)
Rule 2: If \(a\) is any non-zero rational number and \(m, n\) are natural numbers such that \(m>n,\) then \(a^{m} \div a^{n}=a^{(m-n)}\) or \(\frac{\alpha}{a^{4}}=a^{(m-n)}\)
Example: \(\frac{3^{7}}{3^{4}}=\frac{3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3}{3 \times 3 \times 3 \times 3}=3 \times 3 \times 3=3^{3}=3^{(7-4)}\)
Rule 3: If \(a\) is any non-zero rational number and \(m, n\) are natural numbers, then \(\left(a^{m}\right)^{n}=a^{(m \times n)}=\left(a^{n}\right)^{m}\)
Example: \(\left(2^{2}\right)^{3}=2^{2} \times 2^{2} \times 2^{2}=2^{(2+2+2)}=2^{6}=2^{(2 \times 3)}\)
Rule 4: If \(a, b\) are non-zero rational numbers and \(n\) is a natural number, then \(a^{n} \times b^{n}=(a b)^{n}\)
Example: \({2^4} \times {5^4} = (2 \times 2 \times 2 \times 2) \times (5 \times 5 \times 5 \times 5) = (2 \times 5) \times (2 \times 5) \times (2 \times 5) \times (2 \times 5)\)
\( = 10 \times 10 \times 10 \times 10 = {10^4} = {(2 \times 5)^4}\)

Rule 5: If \(a\) and \(b\) are non-zero rational numbers and \(n\) is a natural number, then \(\frac{a^{n}}{b^{n}}=\left(\frac{a}{b}\right)^{n}\)
Example: \(\frac{6^{3}}{5^{3}}=\frac{6 \times 6 \times 6}{5 \times 5 \times 5}=\frac{6}{5} \times \frac{6}{5} \times \frac{6}{5}=\left(\frac{6}{5}\right)^{3}\)

Q.5. What are the types of surds?
Ans: There are three types of surds. They are,
(i) Pure surds
(ii) Mixed surds
(iii) Compound surds

Now you are provided with all the necessary information on exponents and surds and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.