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Chapter 7: Problem 98

Solve each equation. $$x^{-2}=4$$

Short Answer

Expert verified
x = 0.5 or x = -0.5

Step by step solution

01

Rewrite the equation using a positive exponent

Rewrite the equation from \(x^{-2} = 4\) to \(\frac{1}{x^2} = 4\)
02

Isolate the variable

To isolate \(x^2\), take the reciprocal of both sides: \(x^2 = \frac{1}{4}\)
03

Solve for \(x\)

Take the square root of both sides to solve for \(x\):\(x = \frac{1}{2}\) or \(x = -\frac{1}{2}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

negative exponents
Negative exponents might seem tricky at first, but understanding them can simplify equations significantly.
When you see a negative exponent, remember this key idea: it tells you to take the reciprocal (or flip) of the base and change the exponent to a positive one.
For example, with the equation: \( x^{-2} = 4 \), you can rewrite it as \( \frac{1}{x^2} = 4 \).
This transformation makes it easier to isolate and solve for the variable.
square roots
Square roots are fundamental in solving many types of equations, especially quadratic ones.
In our example, after transforming \( x^{-2} = 4 \) to \( \frac{1}{x^2} = 4 \) and then \( x^2 = \frac{1}{4} \), the next step is to take the square root of both sides. This gives us two solutions: \( x = \frac{1}{2} \) and \( x = -\frac{1}{2} \).
Remember, taking the square root of a number always considers positive and negative roots because \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \) and \( \left( -\frac{1}{2} \right)^2 = \frac{1}{4} \) both hold true.
reciprocal
The reciprocal of a number is what you multiply it by to get 1.
For any number \( a \), its reciprocal is \( \frac{1}{a} \). In the context of our equation, changing \( x^{-2} = 4 \) to \( \frac{1}{x^2} = 4 \) made it easier to solve.
Once you have \( x^2 = \frac{1}{4} \), you recognize that taking the square root will lead to the reciprocal values as the solutions. This step ties together the ideas of reciprocal and square roots to find our variable's value.
In conclusion, understanding reciprocals is not just about flipping fractions, it helps bridge steps in solving complex equations.

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