Exponents, Surds, and Indices

Exponents, also known as powers or indices, are a way of representing repeated multiplication of a number by itself. An exponent is usually written as a small superscript number to the right of the base number. For example, 3 raised to the power of 2 is written as 3², which means 3 multiplied by itself twice, or 3 x 3 = 9.

Exponents Rules

Product Rule: When multiplying powers with the same base, keep the base and add the exponents.

For example, a^m x a^n = a^(m+n).

Quotient Rule: When dividing powers with the same base, keep the base and subtract the exponents.

For example, a^m ÷ a^n = a^(m-n).

Power Rule: When raising a power to another power, keep the base and multiply the exponents.

For example, (a^m)^n = a^(m*n).

Zero Rule: Any non-zero number raised to the power of 0 is equal to 1.

For example, a^0 = 1.

Negative Rule: Any non-zero number raised to a negative power is equal to 1 divided by the number raised to the positive power.

For example, a^(-n) = 1/a^n.

Example 1: Evaluate 5³

Solution:

5³ means 5 raised to the power of 3, or 5 multiplied by itself three times: 5 x 5 x 5 = 125. Therefore, 5³ = 125.

Example 2: Simplify (2²)³

Solution:

First, we need to evaluate the expression inside the brackets: 2² = 2 x 2 = 4. Now we can substitute this value into the original expression: (2²)³ = 4³ = 4 x 4 x 4 = 64.

Surds Rules

Simplification Rule: A surd can be simplified if it has a square factor in the radicand.

For example, √50 can be simplified as √(25 x 2) = 5√2.

Addition and Subtraction Rule: Surds can only be added or subtracted if they have the same radicand.

For example, √2 + √2 = 2√2.

Surds are irrational numbers that cannot be expressed exactly as a finite decimal or a fraction. They are often represented by the symbol √, which is called the radical symbol. The number under the radical symbol is called the radicand. For example, √2 is a surd because it is an irrational number.

Example 1: Simplify √18

Solution:

We can simplify √18 by breaking it down into its prime factors: 18 = 2 x 3 x 3. Then, we can rewrite √18 as √(2 x 3 x 3). Using the product rule of surds, we can separate the radical into two parts: √(2 x 3) x √3. Simplifying the first part gives us √6. Therefore, √18 = √6 x √3.

Example 2: Simplify 5√27

Solution:

First, we need to simplify the radical. We can break down 27 into its prime factors: 27 = 3 x 3 x 3. Then, we can rewrite 5√27 as 5 x √(3 x 3 x 3). Using the product rule of surds, we can separate the radical into three parts: √3 x √3 x √3. Simplifying each part gives us 3√3. Therefore, 5√27 = 5 x 3√3 = 15√3.

Indices Rules

Product Rule: When multiplying bases with the same index, keep the index and add the exponents.

For example, a^m x b^m = (ab)^m.

Quotient Rule: When dividing bases with the same index, keep the index and subtract the exponents.

For example, a^m ÷ b^m = (a/b)^m.

Power Rule: When raising a base to another power, keep the base and multiply the index.

For example, (ab)^m = a^m x b^m.

Negative Rule: Any non-zero base raised to a negative index is equal to 1 divided by the base raised to the positive index.

For example, a^(-m) = 1/a^m.

Indices, also known as powers or exponents, are a way of representing repeated multiplication of a number by itself. They are similar to exponents, but instead of using superscript numbers, they use subscript numbers. For example, 2 to the power of 3 is written as 2³, while 2 raised to the index of 3 is written as 2₃.

Example 1: Evaluate 4₃ x 2₄

Solution:

4₃ means 4 raised to the index of 3, or 4 multiplied by itself three times: 4 x 4 x 4 = 64. Similarly, 2₄ means 2 raised to the index of 4, or 2 multiplied by itself four times: 2 x 2 x 2 x 2 = 16. Therefore, 4₃ x 2₄ = 64 x 16 = 1024.

Example 2: Simplify (5x)² ÷ (25x²)

Solution:

First, we need to evaluate the expression inside the brackets: (5x)² = 5x x 5x = 25x². Now we can substitute this value into the original expression: (5x)² ÷ (25x²) = 25x² ÷ (25x²). Using the quotient rule of indices, we can subtract the exponents: 25x² ÷ (25x²) = 1. Therefore, (5x)² ÷ (25x²) simplifies to 1.

Practice Questions on Exponents, Surds, and Indices

Evaluate 2³ x 5².

Solution:

2³ x 5² = 2 x 2 x 2 x 5 x 5 = 8 x 25 = 200.

Simplify √(3² x 2⁴).

Solution:

√(3² x 2⁴) = √(9 x 16) = √144 = 12.

Evaluate 4³ x 2⁶.

Solution:

4³ x 2⁶ = 4 x 4 x 4 x 2 x 2 x 2 x 2 x 2 x 2 = 64 x 64 = 4096.

Simplify √(8x² y³).

Solution:

√(8x² y³) = √(2² x 2 x x² x y³) = 2xy√2y.

Evaluate 3² + 4².

Solution:

3² + 4² = 9 + 16 = 25.

Simplify √(10² + 24²).

Solution:

√(10² + 24²) = √(100 + 576) = √676 = 26.

Evaluate 2⁴ ÷ 2².

Solution:

2⁴ ÷ 2² = 16 ÷ 4 = 4.

Simplify √(3⁴ x 5³).

Solution:

√(3⁴ x 5³) = √(81 x 125) = √10125 = 45.

Evaluate 5⁴.

Solution:

5⁴ = 5 x 5 x 5 x 5 = 625.

Simplify √(16x⁶).

Solution:

√(16x⁶) = √(4² x x⁶) = 4x³.

Evaluate 6² – 3².

Solution:

6² – 3² = 36 – 9 = 27.

Simplify (4² – 3²) ÷ (4 + 3).

Solution:

(4² – 3²) ÷ (4 + 3) = (16 – 9) ÷ 7 = 1

Evaluate (5³)².

Solution:

(5³)² = 5⁶ = 15625

Simplify √(2⁴ x 3²) ÷ √(2² x 9).

Solution:

√(2⁴ x 3²) ÷ √(2² x 9) = 2√3

Evaluate 2⁵ x 3² x 5 ÷ 3³.

Solution:

2⁵ x 3² x 5 ÷ 3³ = 32 x 9 x 5 ÷ 27 = 160 ÷ 27

Simplify 4³ x 5² ÷ 10.

Solution:

4³ x 5² ÷ 10 = 64 x 25 ÷ 10 = 160

Evaluate √(7⁴) ÷ 7².

Solution:

√(7⁴) ÷ 7² = 49 ÷ 49 = 1

Simplify √(27x² y³).

Solution:

√(27x² y³) = √(3³ x x² x y³) = 3xy√3y.

Evaluate 10³ x 2⁴.

Solution:

10³ x 2⁴ = 1000 x 16 = 16000.

Simplify √(5² + 12²).

Solution:

√(5² + 12²) = √(25 + 144) = √169 = 13.

Evaluate 3³ x 2².

Solution:

3³ x 2² = 3 x 3 x 3 x 2 x 2 = 27 x 4 = 108.

Simplify √(3⁶ ÷ 3²).

Solution:

√(3⁶ ÷ 3²) = √(3⁴) = 3² = 9

Evaluate 5⁵ ÷ 5³.

Solution:

5⁵ ÷ 5³ = 5² = 25

Simplify √(5x)² + √(5x)².

Solution:

√(5x)² + √(5x)² = (5x) + (5x) = 10x

Evaluate 2⁴ x 3³ ÷ 6².

Solution:

2⁴ x 3³ ÷ 6² = 16 x 27 ÷ 36 = 12

Simplify √(3x²y⁴) ÷ √(xy²).

Solution:

√(3x²y⁴) ÷ √(xy²) = y√(3y²) ÷ √(x) = y√(3y² ÷ x)

Evaluate (7² + 8²) x 2³.

Solution:

(7² + 8²) x 2³ = (49 + 64) x 8 = 936

Simplify 3⁵ ÷ 3² + 3³.

Solution:

3⁵ ÷ 3² + 3³ = 81 + 27 = 108

Evaluate √(2² + 3²) x 5³.

Solution:

√(2² + 3²) x 5³ = √13 x 125 = 1625√13

Simplify 2⁶ x 5² ÷ 10⁴.

Solution:

2⁶ x 5² ÷ 10⁴ = 64 x 25 ÷ 10000 = 0.16

Evaluate √(2³ x 3²) ÷ √(2² x 3³).

Solution:

√(2³ x 3²) ÷ √(2² x 3³) = 3√2 ÷ 2√3

Simplify √(10⁸) ÷ 10².

Solution:

√(10⁸) ÷ 10² = 10⁴ ÷ 10² = 100

Simplify (a³b²c)² ÷ (abc³)³.

Solution:

(a³b²c)² ÷ (abc³)³ = a⁶b⁴c² ÷ a³b³c⁹ = a³b⁴ ÷ c⁷

Evaluate 5⁴ – 4⁴.

Solution:

5⁴ – 4⁴ = (5² + 4²) (5² – 4²) = 41 x 9 = 369

Simplify 2√(2) x √(8).

Solution:

2√(2) x √(8) = 2√(16) = 8

Evaluate (6⁵ x 9²) ÷ (2⁷ x 3³).

Solution:

(6⁵ x 9²) ÷ (2⁷ x 3³) = (2³ x 3²) x 6² ÷ (2⁷ x 3³) = 27 ÷ 8

Simplify (a² – b²) ÷ (a – b).

Solution:

(a² – b²) ÷ (a – b) = (a + b)(a – b) ÷ (a – b) = a + b

Evaluate (2³ + 3³) ÷ (2² – 2 x 3 + 3²).

Solution:

(2³ + 3³) ÷ (2² – 2 x 3 + 3²) = 35 ÷ 1 = 35

Simplify 3⁵ x 9⁴ ÷ 27².

Solution:

3⁵ x 9⁴ ÷ 27² = 3⁵ x (3²)⁴ ÷ (3²)⁴ = 3⁵

Evaluate √(20² – 16²).

Solution:

√(20² – 16²) = √(336) = 4√21

Simplify 5² x 7⁴ ÷ 35.

Solution:

5² x 7⁴ ÷ 35 = 5² x 7³ = 1225

Simplify √(2 + √3) x √(2 – √3).

Solution:

√(2 + √3) x √(2 – √3) = √(4 – 3) = 1

Evaluate 8² + 6³ ÷ 3².

Solution:

8² + 6³ ÷ 3² = 64 + 72 = 136

Simplify √(25 + 10√6) – √(25 – 10√6).

Solution:

√(25 + 10√6) – √(25 – 10√6) = √[(5 + √6)²] – √[(5 – √6)²] = 5 + √6 – (5 – √6) = 2√6

Evaluate 3⁴ x 7³ ÷ 21².

Solution:

3⁴ x 7³ ÷ 21² = 3⁴ x (7 ÷ 21)² = 3⁴ ÷ 3² = 3² = 9

Simplify (a – b)(a² + ab + b²).

Solution:

(a – b)(a² + ab + b²) = a³ – b³

Evaluate 2³ x (2² + 2 + 1) – 3⁴.

Solution:

2³ x (2² + 2 + 1) – 3⁴ = 8 x 7 – 81 = 7

Simplify (5 + 2√2)².

Solution:

(5 + 2√2)² = 25 + 20√2 + 8 = 33 + 20√2

Evaluate 7³ – 3⁵ + 2⁸ ÷ 2².

Solution:

7³ – 3⁵ + 2⁸ ÷ 2² = 343 – 243 + 64 = 164

Simplify (a⁴ – b⁴) ÷ (a² – b²).

Solution:

(a⁴ – b⁴) ÷ (a² – b²) = (a² + b²)(a² – b²) ÷ (a² – b²) = a² + b²

Evaluate 6³ x 7² ÷ 42.

Solution:

6³ x 7² ÷ 42 = 6³ x (7 ÷ 6)² ÷ 6 = 49