Question

Rewrite the expression as a power of 1/x²:
a) x⁻²
b) (1/x)²
c) x²
d) (x²)⁻¹

Answer 1

All the given expressions
`a) x⁻², b) (1/x)², c) x², and d) (x²)⁻¹` can be rewritten as a power of
`1/x²` by applying the rules for negative exponents and reciprocals.

Step-by-step explanation:

To rewrite the expression as a power of
`1/x²`, we need to convert each of the given expressions accordingly. This involves understanding the rules of exponents particularly regarding negative exponents and how they relate to fractions.

• a)
  `x⁻²`: This expression is already the same as
  `(1/x²)` because a negative exponent indicates the reciprocal. So,
  `x⁻² = 1/x²`.
• b)
  `(1/x)²`: Squaring the fraction
  `1/x` gives us
  `(1²)/(x²)` which is the same as
  `1/x²` since
  `1²` is
  `1`.
• c)
  `x²`: To express
  `x²` as a power of
  `1/x²`, we take the reciprocal and invert the exponent:
  `(x²)⁻¹ = 1/(x²)¹ = 1/x²`.
• d)
  `(x²)⁻¹`: Here, the negative exponent is already present, indicating the reciprocal of
  `x², thus (x²)⁻¹ = 1/x²`.

In conclusion, all the expressions can be rewritten to show the same relationship: they are equal to
`1/x²` either directly or by applying the exponent rules.