/*! 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 33 \(\left(\frac{1}{2}\right)^{-4}\... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 33

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

Short Answer

Expert verified
The solution is 16.

Step by step solution

01

Understand Negative Exponent

A negative exponent signifies that the base (in this case, \(\frac{1}{2}\)) should be inverted and the exponent made positive. Thus \( \left(\frac{1}{2}\right)^{-4} \) becomes equivalent to \( \left(2\right)^{4} \).
02

Apply Positive Exponent

Now, calculate the power of 2 raised to 4. This means multiplying 2 by itself 4 times: \(2^{4} = 2 \times 2 \times 2 \times 2\).
03

Compute the Power

Perform the multiplication: \(2 \times 2 = 4\), then \(4 \times 2 = 8\), and finally \(8 \times 2 = 16\). Thus, \(2^{4} = 16\).

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Key Concepts

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

Exponentiation
Exponentiation refers to the process of raising a base number to the power of an exponent. For instance, in the expression \(2^4\), the number 2 is the base, and 4 is the exponent.
When a number is raised to an exponent, it means you are multiplying the base by itself as many times as the exponent indicates.
For example:
  • \(2^3 = 2 \times 2 \times 2 = 8\)
  • \(5^2 = 5 \times 5 = 25\)
  • \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)
Exponentiation makes large numbers easier to work with by simplifying repeated multiplication.
Base Inversion
Base inversion is a technique used to handle negative exponents. When an exponent is negative, it means you must invert or flip the base before applying the exponent. For example, \(\left(\frac{1}{2}\right)^{-4}\) can be flipped to \((2)^{4}\).
Here's how base inversion works:
  • Find the reciprocal of the base. For \(\left(\frac{1}{2}\right)^{-4}\), the reciprocal is 2.
  • Change the negative exponent to its positive counterpart. So, \(\left(2\right)^{-4}\) becomes \(2^{4}\).
After inverting the base and changing the sign of the exponent, you can then proceed with normal exponentiation.
Power Calculation
Power calculation involves computing the value of a base raised to an exponent. To solve \(2^4\), follow these steps:
1. Multiply 2 by itself 4 times: \(2 \times 2 \times 2 \times 2\).
It can be helpful to do this step-by-step:
  • First, \(2 \times 2 = 4\).
  • Next, \(4 \times 2 = 8\).
  • Finally, \(8 \times 2 = 16\).
So, \(2^4 = 16\). By breaking down the multiplication, it becomes easier to follow and understand each step of the calculation.

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Most popular questions from this chapter

$$ \begin{aligned} &y=-x^{2}+2\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

Find each product. $$ -3 r(r-1)\left(r^{2}+r+1\right) $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ -2 x^{5}\left(3 x^{2}+2 x-5\right)(4 x+2) $$

Find each product. $$ -5 r(r+1)^{3} $$

Perform each indicated operation. \(\left(16 x^{3}-x^{2}+3 x\right)+\left(-12 x^{3}+3 x^{2}+2 x\right)\)
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