Simplify. Express with positive exponents only:
$\left(3 s^7 t^{-5}\right)^2\left(4 s^0 t\right)$
We are given the expression \((3 s^7 t^{-5})^2(4 s^0 t)\). To simplify this, we need to apply the laws of exponents step-by-step. The variables \(s\) and \(t\) are assumed to be non-zero to ensure the expression is well-defined. Our primary goal is to simplify the expression and express it using only positive exponents.
First, we address the term \((3 s^7 t^{-5})^2\). Applying the power of a product rule \((abc)^n = a^n b^n c^n\), we get:
\[(3 s^7 t^{-5})^2 = 3^2 \cdot (s^7)^2 \cdot (t^{-5})^2\]
Using the power of a power rule \((a^m)^n = a^{mn}\):
\[3^2 = 9\]
\[(s^7)^2 = s^{7 \times 2} = s^{14}\]
\[(t^{-5})^2 = t^{-5 \times 2} = t^{-10}\]
So, the simplified first term is \(9 s^{14} t^{-10}\).
Next, we simplify the term \((4 s^0 t)\). We know that any non-zero number raised to the power of zero is 1, so \(s^0 = 1\). Thus:
\[(4 s^0 t) = 4 \cdot 1 \cdot t = 4t\]
Now we combine the two factors by multiplying them together:
\[(9 s^{14} t^{-10}) \cdot (4t) = 9 \cdot 4 \cdot s^{14} \cdot t^{-10} \cdot t^1\]
Multiplying the numerical coefficients gives \(9 \cdot 4 = 36\). For the power of \(t\), we use the product rule \(a^m \cdot a^n = a^{m+n}\):
\[t^{-10} \cdot t^1 = t^{-10+1} = t^{-9}\]
The simplified expression is \(36 s^{14} t^{-9}\).
Finally, we express the result with positive exponents. Using the rule \(a^{-n} = \frac{1}{a^n}\), we rewrite \(t^{-9}\) as \(\frac{1}{t^9}\). Therefore:
\[36 s^{14} t^{-9} = 36 s^{14} \cdot \frac{1}{t^9} = \frac{36 s^{14}}{t^9}\]
The final simplified expression is \(\boxed{\frac{36 s^{14}}{t^9}}\).
Algebraic simplification is essential in fields like physics and engineering, where complex formulas represent physical laws. For instance, when calculating the energy density in an electromagnetic field, you often encounter expressions involving powers of variables like velocity or field strength. Simplifying these expressions allows researchers to interpret the data more efficiently and accurately, ensuring that all physical constraints, such as positive energy or distance, are clearly represented.