$\left(n^2\right)^3=$
$\left(n^2\right)^3=n^\square$
To address the problem, we need to understand the meaning of exponents and the rule for raising a power to a power. The expression \((n^2)^3\) means that the base, which is \(n^2\), is multiplied by itself 3 times. We are asked to write this without using exponents (a) and to find the resulting exponent in the form \(n^x\) (b).
By definition, an exponent tells us how many times to multiply the base by itself. For the expression \((n^2)^3\), the base is \(n^2\). Therefore, we multiply \(n^2\) by itself 3 times:
\[(n^2)^3 = n^2 \times n^2 \times n^2\]
Now, we expand each \(n^2\) term as \(n \times n\):
\[(n^2)^3 = (n \times n) \times (n \times n) \times (n \times n)\]
Counting the factors of \(n\), we have \(n \times n \times n \times n \times n \times n\). Thus, without exponents, the expression is \(n \times n \times n \times n \times n \times n\).
The power rule of exponents states that \((a^m)^n = a^{m \times n}\). Applying this rule to \((n^2)^3\):
\[(n^2)^3 = n^{2 \times 3} = n^6\]
As we saw in the previous step, there are 6 factors of \(n\), which confirms that the resulting exponent is 6. Therefore, \((n^2)^3 = n^6\).
Imagine you are calculating the storage capacity of a computer chip. If each bit is stored as a cell, and you have a 3D grid of cells where each layer is a square of \(n \times n\) cells, you have \(n^2\) cells in one layer. If you have \(n\) such layers, you have \(n^3\) cells. If each cell were replaced by a sub-grid of \(n^2\) smaller components, you would have \((n^2)^3\) components in total. Understanding how exponents interact, like knowing \((n^2)^3 = n^6\), helps in simplifying complex organizational structures in computing and data science.