Final answer:
In mathematics, the power of a product rule lets us multiply exponents when a number is raised to a power multiple times, as in 3²·35 which simplifies to 3³7. For estimations, such as √10 being approximately equal to 3, it's a useful shortcut for calculations. When cubing exponentials, the digit is cubed and the exponent is multiplied by 3; for example, (2³)³ becomes 2³9.
Step-by-step explanation:
Understanding Exponential Operations
When dealing with exponential expressions such as 3²·35, it's important to recognize that this is an application of the law of exponents, specifically the power of a product rule. The expression can be expanded to (3³3) × (3³³3³3³3³3³3), which is equivalent to seven threes multiplied together or 3³7. From this, we can derive the rule: xPxq = x(p+q). This rule states that when you have an exponent raised to another exponent, you multiply the exponents.
Similarly, when discussing approximate values like √10 ~ 3, it means that 3 times 3 is very close to 10, and thus can be used for quick estimations in daily life scenarios, such as budgeting.
When cubing of exponentials, you cube the digit term normally and multiply the exponent by 3. For instance, (2³)³ = 2³9, since the digit term 2 is cubed, resulting in 8, and the exponent 3 is multiplied by 3, yielding 9.
To thoroughly understand these concepts, it's best to connect to the basic concept of integer powers. For example, 10³ is 10 multiplied by itself three times, which gives us 1,000. This foundation in the basics will help to make sense of more complex operations with exponents.