**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.**