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.
