Simplify the expression:
$\left(x^6
ight)^7=x^{\square}$
We are asked to solve for the missing exponent in the expression \((x^6)^7 = x^?\). This problem relies on one of the fundamental rules of exponents, specifically the "power of a power" rule. This rule tells us how to handle cases where a power is raised to another power. It states that for any base \(x\) and exponents \(a\) and \(b\), the rule is \((x^a)^b = x^{a \times b}\).
In the given expression \((x^6)^7\), we can identify the base as \(x\) and the two exponents as \(a = 6\) and \(b = 7\). Applying the power of a power rule directly, we substitute these values into the formula \((x^a)^b = x^{a \times b}\). This gives us:
\[(x^6)^7 = x^{6 \times 7}\]
Now, we simply need to perform the multiplication \(6 \times 7\). Calculating this product, we find:
\[6 \times 7 = 42\]
Thus, the expression becomes \((x^6)^7 = x^{42}\). The missing exponent is \(42\).
The power of a power rule is very useful when simplifying complex algebraic expressions, such as in computer science when calculating how memory capacity scales. For example, if a computer system organizes data in blocks of \(2^{10}\) bytes, and there are \(2^{10}\) such blocks in a larger memory unit, the total memory can be expressed as \((2^{10})^{10}\). Using our rule, we find this equals \(2^{10 \times 10} = 2^{100}\) bytes, helping engineers quickly determine total storage capacity.