To solve the expression [tex]\(\sqrt[3]{27 x^{-3}}\)[/tex], follow these steps:
1. Understand the expression: We need to find the cube root of [tex]\(27 x^{-3}\)[/tex].
2. Simplify the expression inside the cube root:
- We start with [tex]\(27 x^{-3}\)[/tex].
- Recall the property of exponents: [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. Hence, [tex]\(x^{-3} = \frac{1}{x^3}\)[/tex].
- Substituting this back, we get [tex]\(27 x^{-3} = 27 \cdot \frac{1}{x^3} = \frac{27}{x^3}\)[/tex].
3. Find the cube root:
- We now need to find the cube root of [tex]\(\frac{27}{x^3}\)[/tex].
- The cube root of a fraction is the fraction of the cube roots: [tex]\(\sqrt[3]{\frac{27}{x^3}} = \frac{\sqrt[3]{27}}{\sqrt[3]{x^3}}\)[/tex].
4. Calculate the cube roots:
- [tex]\(\sqrt[3]{27} = 3\)[/tex] since [tex]\(3^3 = 27\)[/tex].
- [tex]\(\sqrt[3]{x^3} = x\)[/tex] since the cube root and cube cancel each other out.
5. Put it all together:
- Combining these, we get [tex]\(\frac{3}{x}\)[/tex].
6. Substitute the given value of [tex]\(x\)[/tex]:
- According to the given answer, [tex]\(x = 1\)[/tex].
7. Determine the final result:
- Substituting [tex]\(x = 1\)[/tex], we get [tex]\(\frac{3}{1} = 3\)[/tex].
Therefore, the value of [tex]\(\sqrt[3]{27 x^{-3}}\)[/tex] when [tex]\(x = 1\)[/tex] is [tex]\(3\)[/tex].
