Final answer:
To factor an expression using the greatest common factor and distributive property, identify the GCF of the terms, then use the distributive property to rewrite the expression. In working with exponentials, apply the principles of multiplication, division, and the proper application of exponents. Subtract exponents during division and distribute exponents properly when they're outside parentheses.
Step-by-step explanation:
To use the greatest common factor (GCF) and distributive property to factor expressions, one would first identify the GCF of the terms in the expression. Then, apply the distributive property to rewrite the expression as the GCF multiplied by the simplified terms. For example, consider the expression 8x + 12. The GCF of 8 and 12 is 4, so using the distributive property, the expression can be factored as 4(2x + 3).
When working with exponentials, principles like multiplication or division by the same factor, and the use of the proper power on each term are key to simplification. For instance, (5³)⁴ can be rewritten as 5¹² because when raising a power to a power, you multiply the exponents. This simplification relies on understanding that the distribution of exponents across terms in parentheses is a fundamental property of exponents.
In division of exponentials, divide the coefficient of the numerator by the coefficient of the denominator and subtract the exponent of the denominator from the exponent of the numerator. For example, if you have 2.1× 10^-3³ divided by 7× 10^-2, you would end up with (2.1/7)× 10^-3 - ( -2), which simplifies to 0.3× 10^-1.