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Chapter 4: Problem 12

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) $$ \frac{0^{5}}{2^{0}} $$

Short Answer

Expert verified
The expression evaluates to 0.

Step by step solution

01

Identify the Components of the Expression

First, observe the expression \ \ \frac{0^{5}}{2^{0}}. There are two main components: the numerator (\(0^5\)) and the denominator (\(2^0\)).
02

Calculate the Numerator

The numerator is \(0^5\). Any non-negative number raised to any power is zero. Therefore, \(0^5 = 0\).
03

Calculate the Denominator

The denominator is \(2^0\). Any non-zero number raised to the power of zero equals one. Thus, \(2^0 = 1\).
04

Divide the Numerator by the Denominator

Now, divide the numerator by the denominator: \ \ \frac{0^5}{2^0} = \frac{0}{1}. \ \ Since \(\frac{0}{1} = 0\), the value of the expression is zero.

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Key Concepts

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

Exponentiation Rules
Exponentiation, or raising a number to a power, is a basic concept in algebra. Here, we deal with two types of powers:
  • Zero exponent
  • Non-zero exponent
When you raise any non-zero number to the power of zero, the result is always one. This rule is simple to remember: \[ x^{0} = 1 \] where x is any non-zero number.
For example, in the expression given \[ 2^{0} = 1 \].
On the other hand, raising zero to any positive power always results in zero: \[ 0^{n} = 0 \] where n is any positive integer. For example, in the given problem, \[ 0^{5} = 0 \]. Understanding these basic rules is crucial for simplifying more complex algebraic expressions.
Numerator and Denominator
To fully comprehend the given expression, it's vital to know the roles of the numerator and the denominator in a fraction. The numerator is the top part of the fraction, while the denominator is the bottom part. In the given problem, \[ \frac{0^{5}}{2^{0}} = \frac{0}{1} \], \[ 0^{5} \] is the numerator (0) and \[ 2^{0} \] is the denominator (1).
When simplifying fractions:
  • The numerator tells you how many parts you have.
  • The denominator tells you how many parts make up a whole unit.
Putting it together, a fraction can be simplified by following numerator and denominator rules. In the given example, fractions involving a numerator of zero always evaluate to zero, as \[ \frac{0}{anything} = 0 \]. Thus, simplifying \[ \frac{0}{1} = 0 \].
Division of Zero
Division by zero is an important but sometimes tricky concept in algebra. However, when zero is in the numerator, it's straightforward. \[ \frac{0}{anything} = 0 \], provided 'anything' is a non-zero number.
This rule applies because zero divided by any non-zero number signifies no parts of the denominator's quantity.
In the given problem, \[ \frac{0}{1} = 0 \], confirming that dividing zero by a non-zero number results in zero.
Key things to remember:
  • Zero in the numerator results in zero.
  • Zero in the denominator is undefined (you can't divide by zero).
Understanding these principles helps make sense of algebraic fractions and ensures accurate problem-solving.

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Most popular questions from this chapter

Find each product. $$ \left(9 y+\frac{2}{3}\right)\left(9 y-\frac{2}{3}\right) $$

Our system of numeration is called a decimal system. In a whole number such as 2846 each digit is understood to represent the number of powers of 10 for its place value. The 2 represents two thousands \(\left(2 \times 10^{3}\right),\) the 8 represents eight hundreds \(\left(8 \times 10^{2}\right),\) the 4 represents four tens \(\left(4 \times 10^{1}\right),\) and the 6 represents six ones (or units) \(\left(6 \times 10^{\circ}\right)\) \(2846=\left(2 \times 10^{3}\right)+\left(8 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(6 \times 10^{0}\right) \quad\) Expanded form $$ \text {Keeping this information in mind,} $$ Divide the polynomial \(2 x^{3}+8 x^{2}+4 x+6\) by 2

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ \left(2 x+\frac{2}{3} y\right)\left(3 x-\frac{3}{4} y\right) $$

Use scientific notation to calculate the answer to each problem. Write answers in scientific notation. $$ \frac{650,000,000(0.0000032)}{0.00002} $$

Find each product. $$ (3 x+4 y)(3 x-4 y) $$
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