Simplifying Math with the Laws of Indices: Tips and Tricks

1. The concept of denominator (positive), index, root and exponent, integers, fractions:

Laws of indices are used to simplify expressions with powers or exponents. An exponent is a number that tells you how many times to multiply a base by itself. For example, 2³ means 2 x 2 x 2 = 8. Here are some key terms you should be familiar with:

• Exponent: The exponent is the number that appears as a superscript (above and to the right) of a base number, and it tells you how many times to multiply the base by itself.
• Base: The base is the number that is being multiplied by itself (the number that appears below the exponent).
• Index: An index is the number that tells you what root to take. For example, the square root of 16 has an index of 2.
• Root: A root is the inverse of an exponent. For example, the square root of 16 is 4, because \(4^2\) = 16.
• Integers: Integers are positive and negative whole numbers, including 0. Examples include -3, -2, -1, 0, 1, 2, 3.
• Fractions: Fractions are numbers that represent parts of a whole. They are written as a numerator (top number) over a denominator (bottom number). For example, 1/2 represents one half of a whole.

2. Basic rules of the index and their application:

There are several basic rules of indices that you should be familiar with. Here are some of the most important ones:

• Rule 1: Multiplying powers with the same base - When you multiply two powers with the same base, you can add the exponents. For example, 2³ x 2² = \(2^{(3+2)}=2^5\) = 2^5 = 32.
• Rule 2: Dividing powers with the same base - When you divide two powers with the same base, you can subtract the exponents. For example, 2⁵ ÷ 2³ = \(2^{(5-3)}=2^2\) = 4.
• Rule 3: Powers of powers - When you raise a power to another power, you can multiply the exponents. For example, (2³)² = \(2^{(3\times2)}=2^6\) = 2⁶ = 64.
• Rule 4: Negative exponents - When an exponent is negative, it means that the base is in the denominator. For example, \(2^{(-3)}=1/2^3\) = 1/8.
• Rule 5: Fractional exponents - A fractional exponent is a root. For example, \(4^{(1/2)}\) is the square root of 4, which is 2. Similarly, \(8^{(1/3)}\) is the cube root of 8, which is 2.
• Rule 6: Zero exponents - Anything raised to the power of zero is equal to one. For example, 2⁰ = 1.
• Rule 7: One as an exponent - Anything raised to the power of one is equal to itself. For example, 2¹ = 2

Examples:

Simplify: \((2^2)^5\)
Answer: \(2^{10}\)

Simplify: \(4^2\times4^3\)
Answer: \(4^5\)

Simplify: \((16^2)/(4^2)\)
Answer: \(4^2\)

Simplify: \((3^{(1/4)})^4\)
Answer: \(3^1\)

Simplify: \((8^5)\times (8^2)\)
Answer: \(8^7\)

Simplify: \((4^6)/(4^3)\)
Answer: \(4^3\)

Simplify: \((5^4)\times (5^5)\)
Answer: \(5^9\)

Simplify: \((4^3)^2\)
Answer: \(4^6\)

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