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Chapter 4: Problem 48

Rewrite each expression as simply as you can. $$\left(2 x^{-2}\right)^{2}$$

Short Answer

Expert verified
The simplified expression is \(4 x^{-4}\ \).

Step by step solution

01

Understand the expression

Begin with the given expression: \ \ \ \(\left(2 x^{-2}\right)^{2}\ \). Our goal is to simplify it as much as possible.
02

Apply the power rule to the coefficient

We first focus on \(2^{2}\ \). Applying the power rule, \(2^{2} = 4\ \).
03

Apply the power rule to the variable

Next, handle the exponent for the variable: \(\left(x^{-2}\right)^{2}\ \). According to the power rule \((a^m)^n = a^{m \cdot n}\), multiplying the exponents: \((-2) \cdot 2 = -4\). So, \((x^{-2})^2 = x^{-4}\ \).
04

Combine results

Combining the results from Steps 2 and 3: \ \(4 x^{-4}\ \). This is the simplified expression.

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Key Concepts

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

Power Rule in Algebra
The power rule in algebra states that if you have a base raised to an exponent and then raise the entire expression to another exponent, you multiply the exponents.
That means, \( \big( a^m \big)^n = a^{m \times n} \).
This is a crucial rule for simplifying expressions involving exponents.
For example, if you have \( (2 x^{-2})^2 \), you can simplify it by applying the power rule to both the coefficient and the variable.
The coefficient (2) raised to the power of 2 becomes 4, and the variable exponent is multiplied together to get the new exponent: \( (x^{-2})^2 = x^{-4} \).
This helps in breaking down complex expressions into more manageable forms.
Exponents Simplification
Simplifying exponents involves using different rules to combine and reduce exponents.
As we've used in the problem, the power rule is one such method. Another important rule is the \(a^m \times a^n = a^{m+n}\) rule, which allows the addition of exponents when multiplying like bases.
In the given exercise, after simplifying the exponent using the power rule: \( 2^2 = 4 \) and \( (x^{-2})^2 = x^{-4} \), we combine these results to get the final simplified form: \( 4 x^{-4} \).
Simplification can also involve combining like terms or reducing fractions, but in this example, it primarily focuses on handling exponents accurately.
Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the positive exponent.
For instance, \(a^{-m} = \frac{1}{a^m}\).
This is especially useful for transforming expressions into more usable forms.
In the problem, we have \(x^{-2}\), and when it's squared, it becomes \(x^{-4}\).
This can also be written as \(\frac{1}{x^4}\), but in our simplified result, we've kept the negative exponent.
Understanding and applying rules about negative exponents is key to mastering algebra.
It allows for flexibility in how you present your final answer and ensures thorough simplification of expressions.

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Most popular questions from this chapter

A rock climber launched a hook from the base of a cliff. The edge of the cliff is 100 feet above the climber. Use the formula for the height of a thrown object, \(h=v t-16 t^{2},\) to answer these questions. (Assume that the hook is thrown straight up and, if it travels high enough, catches the edge of the cliff during its descent.) a. Can the hook reach the top of the cliff if its initial velocity is 70 feet per second, or 70 fos? Explain. b. Find an initial velocity that would be sufficient to allow the hook to reach the cliff's edge. c. At an initial velocity of 100 fps, how far above the edge of the cliff will the hook rise? d. At an initial velocity of 100 fps, how long will it take until the hook catches the top of the cliff on its way back down?

Solve each equation by backtracking. (Backtrack mentally if you can.) Check your solutions. $$ 5(n-2)=45 $$

Isabel said it would take her at least an hour and a half, but no more than 2 hours, to finish her homework. Write an inequality to express the number of hours, \(h\), Isabel thinks it will take to do her homework.

The sum of two numbers is 31. One of the numbers is 9 more than the other. a. Write a system of two equations relating the numbers. b. Solve the system to find the numbers.

Graph each inequality. $$y<3 x+7$$
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