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Chapter 7: Problem 3

In Exercises \(1-21,\) solve the equation for the variable. $$ 4=x^{-1 / 2} $$

Short Answer

Expert verified
Answer: \(x=\frac{1}{16}\)

Step by step solution

01

Write down the given equation

The given equation is: $$ 4=x^{-1/2} $$
02

Square both sides of the equation

To get rid of the square root in the denominator of the exponent, square both sides of the equation: $$ (4)^2=(x^{-1/2})^2 $$
03

Simplify the equation

Simplify both sides of the equation: $$ 16=x^{-1} $$
04

Solve for x

Now, we need to get \(x\) out of the exponent. We can do this by taking the reciprocal of both sides of the equation: $$ \frac{1}{16}=x $$ Now, we have solved for \(x\). The solution to the equation is: $$ x=\frac{1}{16} $$

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Key Concepts

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

Exponents and Powers
Exponents and powers play a fundamental role in mathematics by representing repeated multiplication.When we see an expression like \(x^{-1/2}\), it tells us how many times the base (\(x\) in this case) is multiplied by itself.Here, the exponent is \(-1/2\) which might seem confusing, but let's break it down:
  • An exponent of \(-1\) implies taking the reciprocal.
  • A fractional exponent like \(1/2\) refers to the square root.
Combining these concepts, \(x^{-1/2}\) means taking the reciprocal of the square root of \(x\).
A useful rule for exponents to remember is \(x^{a/b} = \sqrt[b]{x^a}\).Just plug in the values, and you have the operation you need!
Reciprocal
Understanding reciprocals is key in algebra, especially when dealing with negative exponents.The reciprocal of a number is simply \(1\) divided by the number itself.For instance, the reciprocal of \(4\) is \(\frac{1}{4}\).
When we see an exponent of \(-1\), as in the modified equation from the solution, \(x^{-1}\), it implies taking the reciprocal of \(x\).That means you flip \(x\) to become \(\frac{1}{x}\).
This operation is often used when solving equations to isolate the variable and simplify expressions.Moreover, applying the reciprocal is consistent with the property that multiplying a number by its reciprocal gives \(1\).
Algebraic Manipulation
Algebraic manipulation is the technique of rearranging equations to isolate and solve for variables.In the given exercise, we apply various manipulations using properties of powers, reciprocals, and arithmetic.
These manipulations often involve one or more of the following:
  • Simplifying expressions
  • Applying the properties of exponents
  • Taking reciprocals
For the equation \((4)^2=(x^{-1/2})^2\), squaring both sides helps eliminate the fractional exponent by multiplying it to become \(-1\).
After simplification, the equation \(16 = x^{-1}\) is solved by applying the reciprocal operation to both sides, leading to \(x = \frac{1}{16}\).Mastering these manipulations is a crucial skill in algebra and allows for the solving of complex equations efficiently.

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Most popular questions from this chapter

Suppose \(c\) is inversely proportional to the square of \(d\). If \(c=50\) when \(d=5\), find the constant of proportion-ality and write the formula for \(c\) in terms of \(d\). Use your formula to find \(c\) when \(d=7\).

In Exercises \(1-4\), write a formula for \(y\) in terms of \(x\) if \(y\) satisfies the given conditions. Proportional to the square of \(x,\) and \(y=1000\) when \(x=5\)

In Problems \(57-59,\) demonstrate a sequence of operations that could be used to solve \(4 x^{2}=16 .\) Begin with the step given. Divide both sides of the equation by \(4 .\)

A cube of side \(x\) has volume \(x^{3} .\) By what factor does the volume change if the length is (a) Doubled? (b) Tripled? (c) Halved? (d) \(\quad\) Multiplied by \(0.1 ?\)

In Exercises \(43-48\), what operation transforms the first equation into the second? Identify any extraneous solutions and any solutions that are lost in the transformation. $$ \begin{aligned} \sqrt{x+4} &=x-2 \\ x+4 &=(x-2)^{2} \end{aligned} $$
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