The Importance of Understanding Laws of Indices in Algebra

1. Understanding the concept of denominator (positive), index, root, and exponent:

Before diving into the laws of indices, it's important to have a solid understanding of some basic concepts related to exponents:

• Exponent: An exponent is a number that tells you how many times to multiply a base number by itself. For example, in the expression \(3^4\), 3 is the base and 4 is the exponent. This means you should multiply 3 by itself 4 times: 3 x 3 x 3 x 3 = 81.
• Root: A root is the opposite of an exponent. It tells you what number must be multiplied by itself a certain number of times to get the original number. For example, the square root of 16 is 4 because 4 x 4 = 16.
• Index: An index is a small number written to the top left of a root symbol. It tells you what kind of root to take. For example, the cube root of 27 can be written as \(27^{(1/3)}\), where 1/3 is the index.
• Denominator: The denominator is the number at the bottom of a fraction. In the context of indices, a positive denominator is used to represent a root, such as a square root or cube root.

It's important to note that exponents can be positive or negative, and can be integers or fractions. Positive exponents indicate repeated multiplication, while negative exponents indicate division by the base raised to the positive exponent.

2. Basic rules of the index and their application:

Now that we have a solid understanding of the basic concepts related to exponents, we can move on to the laws of indices. These are rules that help simplify expressions with exponents and make calculations easier. Here are the basic rules of indices:

• Product rule: When multiplying two numbers with the same base, add the exponents. For example, \(2^3\times 2^4=2^{(3+4)}=2^7\).
• Quotient rule: When dividing two numbers with the same base, subtract the exponents. For example, \(5^6/5^3=5^{(6-3)}=5^3\).
• Power rule: When raising a power to another power, multiply the exponents. For example, \((2^3)^4=2^{(3\times4)}=2^{(12)}\).
• Negative rule: When a negative exponent is present, move the base to the denominator and change the exponent to positive. For example, \(4^{(-2)}=1/4^2\).
• Fractional rule: When a fraction exponent is present, take the root of the base and raise it to the numerator, and then take the root of the result and raise it to the denominator. For example, \(16^{(1/2)}=\sqrt16\) = 4, and \(4^{(1/3)}=\sqrt[3] {4}\).

Examples:

Simplify the expression \(2^3\times 2^4\).
Answer: We can use the product rule to add the exponents and get \(2^{(3+4)}=2^7\). Therefore, the simplified expression is 128.

Evaluate the expression \((3^2)^3\).
Answer: We can use the power rule to multiply the exponents and get \(3^{(2\times3)}=3^6\). Therefore, the value of the expression is 729.

Simplify the expression \(10^3/10^5\).
Answer: We can use the quotient rule to subtract the exponents and get \(10^{(3-5)}=10^{(-2)}\). Since this is a negative exponent, we can move the base to the denominator and change the exponent to positive to get \(1/10^2\) = 1/100. Therefore, the simplified expression is 1/100.

Evaluate the expression \(5^{(-3)}\).
Answer: Since this is a negative exponent, we can move the base to the denominator and change the exponent to positive to get \(1/5^3\). Therefore, the value of the expression is 1/125.

What is the product rule of indices, and how is it used to simplify expressions?
Answer: Simplify the expression \(5^2\times5^4\) using the product rule of indices.

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