Final answer:
To use properties of exponents to express an equivalent expression with a single exponent, you multiply the digit terms and add the exponents of like bases when multiplying exponentials, square the digit term and multiply the exponent by 2 when squaring, and cube the digit term and multiply the exponent by 3 when cubing.
Step-by-step explanation:
When we use the properties of exponents, we can simplify expressions with exponents in a more concise form. For multiplication of exponentials, we can add the exponents together when the bases of the expressions being multiplied are the same. For instance, looking at Example A.7.1, we have 3.2 × 10³ times 2 × 10² written as 3.2 · 2 · 10³ · 10², simplifying to 6.4 × 10⁵ by multiplying the digit terms and adding the exponents of the common base 10.
We also apply similar logic with other operations, such as squaring of exponentials and cubing of exponentials. When squaring an exponential term like (5³)², we square the digit term and multiply the exponent by 2, resulting in 5⁴ (or 5 raised to the 12th power), as we effectively have twelve 5s multiplied together. It is crucial to remember that we can only add or multiply the exponents when the bases are the same, which will not work if the bases are different, as shown in A.8.