Final answer:
Laws of exponents allow us to perform operations on numbers with powers. They include adding exponents when multiplying with the same base and multiplying exponents when raising a power to a power. These laws are essential for working with scientific notation and simplifying algebraic expressions.
Step-by-step explanation:
Understanding the laws of exponents is crucial for solving problems involving powers of numbers. The laws allow us to manipulate exponential expressions effectively and can be applied in various scenarios, including scientific notation and algebraic simplifications.
Applying Laws of Exponents
For example, when multiplying numbers in scientific notation, as shown in example 3.2 × 10³ times 2 × 10², we apply the law stating that when multiplying powers with the same base, we add the exponents. First, we multiply the coefficients: 3.2 × 2 = 6.4. Then we add the exponents for the base 10: 3 + 2 = 5. The product is then written as 6.4 × 10⁵.
Another law of exponents is when raising a power to a power, such as (5³)⁴. In this case, we multiply the exponents together: 3 × 4 = 12, resulting in 5¹². Similarly, if we have two variables with negative exponents while multiplying, for instance, 2.4 x 10⁻², we can recognize that moving the decimal place two spots to the left results in 0.024, applying the rule for negative exponents and scientific notation.