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Chapter 5: Problem 7

Complete each table. \(\begin{array}{|r|r|}\hline x & {3^{x}} \\ \hline 2 & {} \\ \hline 1 & {} \\\ \hline 0 & {} \\ \hline-1 & {} \\ \hline-2 & {} \\ \hline\end{array}\)

Short Answer

Expert verified
9, 3, 1, \(\frac{1}{3}\), \(\frac{1}{9}\).

Step by step solution

01

Calculate for x = 2

To find the value of \(3^x\) when \(x = 2\), calculate \(3^2\). This is done by multiplying 3 by itself: \(3 \times 3 = 9\). So, the value is 9.
02

Calculate for x = 1

To find the value of \(3^x\) when \(x = 1\), calculate \(3^1\). Since any number to the power of 1 is itself, \(3^1 = 3\). The value is 3.
03

Calculate for x = 0

To find the value of \(3^x\) when \(x = 0\), calculate \(3^0\). Any number raised to the power of 0 is 1. Therefore, \(3^0 = 1\). The value is 1.
04

Calculate for x = -1

To find the value of \(3^x\) when \(x = -1\), calculate \(3^{-1}\). This is the reciprocal of \(3^1\), so \(3^{-1} = \frac{1}{3}\). The value is \(\frac{1}{3}\).
05

Calculate for x = -2

To find the value of \(3^x\) when \(x = -2\), calculate \(3^{-2}\). This is the reciprocal of \(3^2\), so \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\). The value is \(\frac{1}{9}\).

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Key Concepts

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

Understanding Powers of Numbers
The concept of powers of numbers involves multiplying a number by itself a specified number of times. This is represented by the expression \(a^n\), where \(a\) is the base, and \(n\) is the exponent, which tells us how many times to use the base in a multiplication. For example, \(3^2\) means 3 multiplied by itself: \(3 \times 3 = 9\). Here are some simple rules:
  • If the exponent is 1, the number remains unchanged: \(3^1 = 3\).
  • If the exponent is 0, the result is always 1, regardless of the base: \(3^0 = 1\).
  • Each increase in the exponent is like adding another multiplication: \(3^3 = 3 \times 3 \times 3\).
These rules help you quickly identify the result of a number raised to a certain power.
Exploring Negative Exponents
Negative exponents might seem tricky at first, but they represent a simple concept: the reciprocal of the positive exponent. When you see an expression like \(3^{-1}\), this is equal to the reciprocal of \(3^1\). The reciprocal of a number is 1 divided by that number.For example:
  • \(3^{-1} = \frac{1}{3}\)
  • \(3^{-2}\) means we take \(3^2\) and find its reciprocal, so \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)
Whenever you encounter a negative exponent, think of flipping the fraction. This means using the base's inverse while keeping the exponent in positive terms. The negative exponent indicates that the result will be a fraction.
Reciprocals Simplified
Reciprocals are a fundamental concept when dealing with fractions and exponents. The reciprocal of a number is essentially what you multiply by the original number to get 1. For example, the reciprocal of 3 is \(\frac{1}{3}\), since:
  • \(3 \times \frac{1}{3} = 1\)
When working with exponents, especially negative ones, switching the base to its reciprocal can solve a problem quickly. Say you have \(2^{鈭�1}\), to find it, you would take the reciprocal of \(2^1\), resulting in \(\frac{1}{2}\). The rules of reciprocals help simplify expressions and solve equations involving negative exponents.
Basics of Multiplying Numbers
Multiplication is one of the simplest arithmetic operations. When multiplying numbers, you're essentially adding a number to itself a specified number of times. For example, when you multiply 3 by 2, it's the same as adding 3 two times: \(3 + 3 = 6\). When it comes to exponents, such as \(3^2\), this means you multiply the base (3) by itself (3) once more, ending in the product 9.Here's a quick reminder of some multiplication basics:
  • When multiplying more than two numbers, multiply two numbers at a time: \(2 \times 3 \times 4 = (2 \times 3) \times 4 = 6 \times 4 = 24\).
  • Multiplying any number by 1 leaves the original number unchanged: \(5 \times 1 = 5\).
  • Multiplying by 0 results in zero: \(7 \times 0 = 0\).
Understanding multiplication is crucial for interpreting powers, as it frequently involves multiplying the same number multiple times.

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