Final answer:
Equivalent expressions in algebra involving radicals and rational exponents are simplified by applying exponent rules, such as the power rule, product rule, and quotient rule, which streamline the process of multiplying, dividing, and converting between different forms of exponents.
Step-by-step explanation:
Equivalent expressions are created in algebra to help simplify expressions involving radicals and rational exponents by applying a series of exponent rules. These rules, which are pivotal in simplifying mathematical expressions, include the power rule, product rule, quotient rule, and the conversion between radicals and exponents. For instance, if we take an expression like 5¹ · 5¹, we can apply the product rule of exponents and add the exponents, resulting in 5¹, which simplifies to 5. This is because 5 is considered the square root of 25, equivalent to 5². Hence, we can represent the square root of a number x using a fractional exponent as x¹⁄₂ (√x).
When multiplying two exponentiated quantities, such as (5³)⁴, we multiply the exponents (3 × 4) to achieve 5¹₂. Similarly, when dividing exponential terms, we subtract the exponents. For instance, 10³ ÷ 10² simplifies to 10¹ by subtracting the exponents. Multiplying scientific notations is also simplified by multiplying the numbers out front and adding the exponents of powers of ten, as shown in the calculation (3 × 10⁵) × (2 × 10°) = 6 × 10⁵.