Laws of Exponents: Positive and Negative Integral Exponents, and Zero Exponents (Filipino)

Summary

This tutorial covers the foundational concepts of exponents, including integral exponents, zero exponents, and the laws governing positive and negative exponents. It explains how exponents represent repeated multiplication and provides examples of simplifying expressions with powers. Key rules discussed include raising numbers to zero or negative exponents, multiplying or dividing expressions with the same base, and applying exponents to products or fractions. The tutorial also addresses common mistakes and illustrates the process of simplifying complex algebraic expressions using these exponent laws.

Takeaways

• 😀 Exponents represent repeated multiplication of a number by itself, with the exponent indicating how many times the base number is multiplied.
• 😀 Integral exponents are whole numbers (positive, negative, or zero) that are used to simplify expressions in algebra.
• 😀 A negative exponent indicates the reciprocal of the base raised to the positive exponent, e.g., x^(-n) = 1/x^n.
• 😀 Any non-zero number raised to the zero power equals 1, e.g., 4^0 = 1.
• 😀 A number raised to the first power is the number itself, e.g., a^1 = a.
• 😀 When multiplying powers with the same base, add the exponents: x^m * x^n = x^(m+n).
• 😀 When dividing powers with the same base, subtract the exponents: x^m / x^n = x^(m-n).
• 😀 Raising a power to another power involves multiplying the exponents: (x^m)^n = x^(m*n).
• 😀 The product rule of exponents states that when you raise a product to a power, apply the exponent to each factor: (ab)^n = a^n * b^n.
• 😀 The quotient rule of exponents states that when you raise a quotient to a power, apply the exponent to both the numerator and the denominator: (a/b)^n = a^n / b^n.

Q & A

What is an exponent?

An exponent indicates how many times a number (the base) is multiplied by itself. For example, in 3^4, the exponent '4' means that 3 is multiplied by itself four times: 3 * 3 * 3 * 3.

What does a negative exponent indicate?

A negative exponent means you take the reciprocal of the base raised to the positive exponent. For example, x^(-4) is the same as 1 / x^4.

How do you simplify expressions with zero exponents?

Any non-zero number raised to the zero power is equal to 1. For instance, 4^0 = 1 and x^0 = 1, as long as x is not equal to zero.

What happens when you raise a number to the first power?

When a number is raised to the first power, it stays the same. For example, 5^1 = 5 and x^1 = x.

How do you handle multiplying expressions with the same base but different exponents?

When multiplying expressions with the same base, you add the exponents. For example, x^2 * x^4 = x^(2+4) = x^6.

How do you divide expressions with the same base but different exponents?

When dividing expressions with the same base, you subtract the exponents. For example, y^7 / y^4 = y^(7-4) = y^3.

What is the rule for raising a product to an exponent?

When raising a product to an exponent, apply the exponent to each factor in the product. For example, (2y)^4 = 2^4 * y^4 = 16y^4.

How do you simplify expressions where the base is a fraction and the exponent is a positive integer?

When raising a fraction to a positive exponent, raise both the numerator and denominator to that exponent. For example, (x/3)^2 = x^2 / 3^2 = x^2 / 9.

What is the result when you raise a negative number to an even exponent?

When a negative number is raised to an even exponent, the result is positive. For example, (-5)^2 = 25.

How do you handle raising a power to another power, such as (x^5)^3?

When raising a power to another power, you multiply the exponents. For example, (x^5)^3 = x^(5*3) = x^15.