Which expression is equivalent to $\left(4 h^7 k^2\right)^4$?
To simplify the expression \((4 h^7 k^2)^4\), we use the laws of exponents. These laws are essential for manipulating algebraic expressions involving powers. Specifically, we will use the "power of a product" rule and the "power of a power" rule. The power of a product rule states that \((abc)^n = a^n \cdot b^n \cdot c^n\). The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). These rules allow us to break down the expression into smaller, manageable parts.
Applying the power of a product rule, we distribute the exponent \(4\) to each factor inside the parentheses:
\[(4 h^7 k^2)^4 = 4^4 \cdot (h^7)^4 \cdot (k^2)^4.\]
Now, we calculate the value of each part separately. First, for the constant term, we find \(4^4 = 4 \times 4 \times 4 \times 4 = 16 \times 16 = 256\). Next, we apply the power of a power rule to the variable terms. For \(h^7\), we get \((h^7)^4 = h^{7 \times 4} = h^{28}\). For \(k^2\), we get \((k^2)^4 = k^{2 \times 4} = k^8\).
Combining all the calculated parts together, we get:
\[256 \cdot h^{28} \cdot k^8 = 256 h^{28} k^8.\]
This final expression is equivalent to the original one \((4 h^7 k^2)^4\). By comparing this result with the given choices, we see that it corresponds to the fourth option: \(256 h^{28} k^8\).
Exponents are frequently used to understand how physical quantities change when dimensions are scaled. For example, if you have a cube with side length \(s\), its volume is \(V = s^3\). If you decide to triple the side length, the new side length becomes \(3s\). The new volume is \((3s)^3 = 3^3 \cdot s^3 = 27s^3\). This shows that by simply tripling the side length, the volume increases by a factor of 27. Understanding exponent rules allows us to calculate how scaling affects area, volume, and many other real-world measurements efficiently!