/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 \(\left(\frac{1}{4}\right)^{-2}\... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 5

\(\left(\frac{1}{4}\right)^{-2}\)

Short Answer

Expert verified
16

Step by step solution

01

Understand the problem

The exercise requires simplifying the expression \(\frac{1}{4}\right)^{-2}\). Note that when we have a negative exponent, it means we take the reciprocal of the base.
02

Apply the negative exponent rule

To handle the negative exponent, take the reciprocal of \(\frac{1}{4}\right)\). Therefore, \(\frac{1}{4}\right)^{-2}\) becomes \(\frac{4}{1}\right)^{2}\). This simplifies to \(4^2\).
03

Calculate the power

Now, calculate the square of 4. \(4^2\) is equal to 16.
04

Final Answer

The result after simplifying \(\frac{1}{4}\right)^{-2}\) is 16.

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Key Concepts

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

Reciprocal
In mathematics, the reciprocal of a number is simply the number flipped upside down. For a fraction, like \(\frac{1}{4}\), the reciprocal is \(\frac{4}{1}\).

When dealing with negative exponents, understanding reciprocals is crucial. A negative exponent in an expression indicates that you need to take the reciprocal of the base and then raise it to the positive version of the exponent.

For example, \(\frac{1}{4}^{-2}\) tells us to first find the reciprocal of \(\frac{1}{4}\), which is \(\frac{4}{1}\), and then raise it to the power of 2. Knowing this concept makes understanding negative exponents much easier.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This often requires understanding and applying rules of arithmetic and algebra.

In our example \(\left(\frac{1}{4}\right)^{-2}\), the negative exponent tells us to take the reciprocal first. So, \(\frac{1}{4}^{-2}\) simplifies to \(\frac{4}{1}^{2}\)\.

Once you have the reciprocal, the next step is to calculate the power. In this case, \(\frac{4}{1}\) becomes 4, and raising 4 to the power of 2 is straightforward.
Power Calculations
Power calculations involve raising a number or a base to an exponent. The exponent indicates how many times the base is multiplied by itself.

For instance, in \(4^{2}\), the base is 4, and the exponent is 2. This means 4 times 4, which equals 16.

In our original expression \(\left(\frac{1}{4}\right)^{-2}\), after taking the reciprocal to get \(\frac{4}{1}^{2}\), we need to compute 4 raised to the power of 2, resulting in 16.

To sum up, mastering power calculations helps you tackle many algebraic problems with ease.

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Most popular questions from this chapter

Find each product. $$ -5 t^{4}\left(2 t^{4}+1\right)\left(2 t^{4}-1\right) $$

Add or subtract as indicated. \(\left(3 k^{2} h^{3}+5 k h+6 k^{3} h^{2}\right)-\left(2 k^{2} h^{3}-9 k h+k^{3} h^{2}\right)\)

Perform each indicated operation. \(\left(8 m^{2}-7 m\right)-\left(3 m^{2}+7 m-6\right)\)

Perform each indicated operation. \(\left[\left(3 x^{2}-2 x+7\right)-\left(4 x^{2}+2 x-3\right)\right]-\left[\left(9 x^{2}+4 x-6\right)+\left(-4 x^{2}+4 x+4\right)\right]\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 5(3 k-7)(5 k+2) $$
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