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Chapter 4: Problem 13

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) $$ 8^{0}-12^{0} $$

Short Answer

Expert verified
The expression is equal to 0.

Step by step solution

01

Understand Exponentiation by Zero

Recall the rule for any non-zero number raised to the power of zero: Any number raised to the power of zero is equal to 1. Mathematically, this is represented as: \[a^0 = 1\] for any non-zero number \(a\).
02

Apply the Rule to Each Term

Apply the rule to both terms in the expression:\[8^0 = 1\] and\[12^0 = 1\].
03

Perform the Subtraction

Subtract the two results obtained from Step 2:\[1 - 1 = 0\].
04

State the Answer

The expression \[8^0 - 12^0\] simplifies to 0. Thus, the final answer is 0.

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Key Concepts

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

Zero Exponent Rule
One of the most important rules in mathematics is the Zero Exponent Rule. This rule states that any non-zero number raised to the power of zero equals 1. For example,
  • If you have 5 raised to the power of zero, it is written as:
    5^0 = 1
  • Similarly, for any number 'a,' as long as 'a' is not zero, a^0 = 1.
This rule is very useful while simplifying expressions, as it can greatly reduce the complexity of the expression.

If you look at our exercise, we used the Zero Exponent Rule on both 8 and 12 to transform them into simpler terms:
  • 8^0 = 1
  • 12^0 = 1
This step made it easy to perform further calculations.
Exponentiation
Understanding exponentiation is crucial in mathematics.

Exponentiation involves raising a number (the base) to the power of an exponent. For example,
  • 3^2 means 3 raised to the power of 2, which equals 9.
The process of raising a number to an exponent simplifies repeated multiplication. Let鈥檚 look at the expression from our exercise:
  • 8^0 - 12^0
Applying exponentiation:
  • 8^0 results in 1 (based on the Zero Exponent Rule).
  • 12^0 results in 1 (again, based on the Zero Exponent Rule).
    • When you apply exponentiation properly and consistently, the expression simplifies readily.
Simplifying Expressions
Simplifying expressions means making them easier to understand or work with by reducing their complexity.

In our exercise, we began with the expression 8^0 - 12^0. By applying the Zero Exponent Rule, both terms simplified to 1, resulting in the simpler expression 1-1.

Once simplified, you can easily perform the operation to get the result:
  • 1 - 1 = 0.
The final answer after simplification is 0.
Breaking down expressions into simpler parts and then simplifying each part step-by-step makes it easier to solve complex problems.

Always break down the expression:
  • Identify parts that can be simplified.
  • Apply rules such as the Zero Exponent Rule.
  • Perform the arithmetic operations.
These steps help in solving and simplifying even the most complicated expressions.

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Most popular questions from this chapter

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ can be used to perform some multiplication problems. Here are two examples. $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1 \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 103 \times 97 $$

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 2846 by \(2,\) using paper-and-pencil methods: \(2 \longdiv { 2 8 4 6 }\)

Evaluate each expression for \(x=3 .\) See Sections 1.3 and 1.6. $$ 2 x^{2}-3 x+10 $$

A polynomial in the variable \(x\) has degree 6 and is divided by a monomial in the variable \(x\) having degree \(4 .\) What is the degree of the quotient?

Evaluate. $$ 49 \div 10,000 $$
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