Problem 3 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

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

Step 1: Write down the given equation

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

Step 2: 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: $$ (4)^2=(x^{-1/2})^2 $$

Step 3: Simplify the equation

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

Step 4: 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: $$ x=\frac{1}{16} $$

Key Concepts

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.