Sure, let's solve the equation [tex]\(\square^{-3} = 27\)[/tex] step-by-step.
1. Understanding the given equation:
The equation is [tex]\(\square^{-3} = 27\)[/tex]. Here, [tex]\(\square\)[/tex] represents an unknown value that we need to determine.
2. Rewrite the equation:
We need to solve for [tex]\(\square\)[/tex]. Let's denote the unknown value by [tex]\(a\)[/tex]. Thus, our equation becomes:
[tex]\[
a^{-3} = 27
\][/tex]
3. Rewrite [tex]\(a^{-3}\)[/tex] in terms of positive exponents:
The term [tex]\(a^{-3}\)[/tex] can be written as:
[tex]\[
\frac{1}{a^3} = 27
\][/tex]
4. Isolate [tex]\(a^3\)[/tex]:
To isolate [tex]\(a^3\)[/tex], we take the reciprocal of both sides of the equation. This gives us:
[tex]\[
a^3 = \frac{1}{27}
\][/tex]
5. Solve for [tex]\(a\)[/tex]:
To solve for [tex]\(a\)[/tex], we need to take the cube root on both sides of the equation. Taking the cube root yields:
[tex]\[
a = \sqrt[3]{\frac{1}{27}}
\][/tex]
6. Simplify the cube root:
We know that [tex]\(\frac{1}{27}\)[/tex] can also be written as [tex]\(27^{-1}\)[/tex]. Therefore:
[tex]\[
a = \left(27^{-1}\right)^{\frac{1}{3}}
\][/tex]
Using the power rule of exponents—[tex]\(\left(x^m\right)^n = x^{mn}\)[/tex]—we combine the exponents:
[tex]\[
a = 27^{-\frac{1}{3}}
\][/tex]
7. Calculate the final value:
By evaluating [tex]\(27^{-\frac{1}{3}}\)[/tex], we find:
[tex]\[
a \approx 0.33333333333333337
\][/tex]
So, the value of [tex]\(\square\)[/tex] (or [tex]\(a\)[/tex]) is approximately [tex]\(\boxed{0.33333333333333337}\)[/tex].
