Surds and Indices Key Concepts, Simplifications, and Practice Questions

Surds Questions:

1. Simplify 48\sqrt{48}48​.
   • Answer:
     $\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$48​=16×3​=16​×3​=43​.
2. Simplify 35+253\sqrt{5} + 2\sqrt{5}35​+25​.
   • Answer:
     $3\sqrt{5} + 2\sqrt{5} = (3 + 2)\sqrt{5} = 5\sqrt{5}$35​+25​=(3+2)5​=55​.
3. Simplify 205\frac{\sqrt{20}}{\sqrt{5}}5​20​​.
   • Answer:
     $\frac{\sqrt{20}}{\sqrt{5}} = \sqrt{\frac{20}{5}} = \sqrt{4} = 2$5​20​​=520​​=4​=2.
4. Rationalize the denominator of 32\frac{3}{\sqrt{2}}2​3​.
   • Answer:
     $\frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}$2​3​×2​2​​=232​​.
5. Simplify 75−27\sqrt{75} – \sqrt{27}75​−27​.
   • Answer:
     $\sqrt{75} – \sqrt{27} = \sqrt{25 \times 3} – \sqrt{9 \times 3} = 5\sqrt{3} – 3\sqrt{3} = (5 – 3)\sqrt{3} = 2\sqrt{3}$75​−27​=25×3​−9×3​=53​−33​=(5−3)3​=23​.

Indices Questions:

Answer:
$16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \times \frac{3}{4}} = 2^3 = 8$1643​=(24)43​=24×43​=23=8.

Simplify 24×232^4 \times 2^324×23.

Answer:
$2^4 \times 2^3 = 2^{4+3} = 2^7 = 128$24×23=24+3=27=128.

Simplify 3532\frac{3^5}{3^2}3235​.

Answer:
$\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$3235​=35−2=33=27.

Simplify (22)3(2^2)^3(22)3.

Answer:
$(2^2)^3 = 2^{2 \times 3} = 2^6 = 64$(22)3=22×3=26=64.

Simplify 5−25^{-2}5−2.

Answer:
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$5−2=521​=251​.

Simplify 163416^{\frac{3}{4}}1643​.

Surds and Indices: Key Concepts and Applications

Surds and indices are two essential topics in algebra, and understanding them is crucial for solving many types of problems in mathematics, especially in competitive exams.

1. Surds:

Definition:
A surd is an expression that contains a square root, cube root, or any other root that cannot be simplified into a rational number.

Key Points about Surds:

• A number or expression in a square root that cannot be simplified into a whole number is called a surd.
• Surds can be left in their root form unless there’s a way to simplify them.
• For example, $\sqrt{2}$2​ is a surd, but $\sqrt{4} = 2$4​=2, which is not a surd.

Simplifying Surds:

• Simplify surds by factoring the number inside the root and taking out any square (or higher powers) as whole numbers.
• Example: $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$18​=9×2​=9​×2​=32​.

Operations with Surds:

• Addition/Subtraction:
  • You can only add or subtract surds that are of the same form (like terms).
  • Example: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$32​+52​=82​.
• Multiplication:
  • When multiplying surds, multiply the numbers inside the root first, and then take the root.
  • Example: $\sqrt{3} \times \sqrt{5} = \sqrt{15}$3​×5​=15​.
• Division:
  • When dividing surds, divide the numbers inside the roots.
  • Example: $\frac{\sqrt{8}}{\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2$2​8​​=28​​=4​=2.

Rationalizing Surds:

• If there’s a surd in the denominator, multiply both the numerator and denominator by the conjugate to remove the surd.
• Example: $\frac{1}{\sqrt{2}}$2​1​ becomes $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$2​1​×2​2​​=22​​.

2. Indices (Exponents):

Definition:
An index or exponent is the number that shows how many times the base number is multiplied by itself.

For example, in $2^3$23, the base is 2, and the index is 3, meaning $2 \times 2 \times 2 = 8$2×2×2=8.

Laws of Indices:

1. Multiplying Powers with the Same Base:$$a^m \times a^n = a^{m + n}$$am×an=am+n
   • Example: $2^3 \times 2^4 = 2^{3+4} = 2^7$23×24=23+4=27.
2. Dividing Powers with the Same Base:$$\frac{a^m}{a^n} = a^{m – n}$$anam​=am−n
   • Example: $\frac{2^5}{2^2} = 2^{5-2} = 2^3$2225​=25−2=23.
3. Power of a Power:$$(a^m)^n = a^{m \times n}$$(am)n=am×n
   • Example: $(2^3)^2 = 2^{3 \times 2} = 2^6$(23)2=23×2=26.
4. Multiplying Powers with Different Bases but the Same Exponent:$$a^m \times b^m = (a \times b)^m$$am×bm=(a×b)m
   • Example: $2^3 \times 3^3 = (2 \times 3)^3 = 6^3$23×33=(2×3)3=63.
5. Dividing Powers with Different Bases but the Same Exponent:$$\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$$bmam​=(ba​)m
   • Example: $\frac{2^4}{3^4} = \left(\frac{2}{3}\right)^4$3424​=(32​)4.
6. Negative Exponent:$$a^{-m} = \frac{1}{a^m}$$a−m=am1​
   • Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$2−3=231​=81​.
7. Zero Exponent:$$a^0 = 1 \quad \text{(for any non-zero a)}$$a0=1(for any non-zero a)
   • Example: $5^0 = 1$50=1.
8. Fractional Exponent:$$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$anm​=nam​
   • Example: $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$831​=38​=2.

Key Formulas and Concepts:

$a^{-m} = \frac{1}{a^m}$a−m=am1​

Simplifying Surds:

$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$a​×b​=a×b​

$\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}$a​÷b​=ba​​

Rationalizing the Denominator:

$\frac{1}{\sqrt{a}}$a​1​ becomes $\frac{\sqrt{a}}{a}$aa​​.

$\frac{1}{a + \sqrt{b}}$a+b​1​ becomes $\frac{a – \sqrt{b}}{a^2 – b}$a2−ba−b​​ (conjugate method).

Laws of Indices:

$a^m \times a^n = a^{m+n}$am×an=am+n

$\frac{a^m}{a^n} = a^{m-n}$anam​=am−n

$(a^m)^n = a^{mn}$(am)n=amn