Final answer:
The simplified expression of (m³n⁴)² is m⁶n⁸, where we multiply the individual exponents by the outer exponent. This applies to variables just like it does for numbers or unit measures in dimensional analysis.
Step-by-step explanation:
To simplify the expression (m³n⁴)², we apply the power of a power rule, which states that (a⁴²) = a⁴²×². For our expression, we multiply the exponents inside the parentheses by the exponent outside the parentheses:
- m³ raised to the 2nd power becomes m³×² = m⁶
- n⁴ raised to the 2nd power becomes n⁴ײ = n⁸
Therefore, the simplified expression is m⁶n⁸.
When dealing with exponentiation and dimensional analysis, it's essential to apply the rules consistently, whether the base is a variable, a number, or a unit of measure. For example:
- When you have an expression like (27x³)(4x²), it equals 2.1 × 10⁻³³.
- For converting square centimetres to square meters, remember that 1 cm² = (10⁻ m)² = 10⁻⁴ m².
It is also helpful to break down expressions into their base components when converting units or working with derived units like m².