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Chapter 4: Problem 1

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) $$ 9^{0} $$

Short Answer

Expert verified
The expression \(9^0\) equals 1.

Step by step solution

01

Understand the zero exponent rule

According to the zero exponent rule, any non-zero number raised to the power of 0 is equal to 1.
02

Apply the zero exponent rule to the expression

Using the zero exponent rule, apply it to the given expression: \(9^0 = 1 \)

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental concept in mathematics. They represent how many times a number, called the base, is multiplied by itself. For example, in the expression \(3^4\), the base is 3 and the exponent is 4, which means \(3 \times 3 \times 3 \times 3\).

Exponents make it easier to write and work with large numbers and repeated multiplications. Here are some key points to remember:
  • The exponent tells us how many times to multiply the base by itself.
  • Positive exponents indicate repeated multiplication.
  • Negative exponents indicate repeated division or taking the reciprocal.
  • The zero exponent, which we'll discuss in detail, has its own special rule.
The Zero Exponent Rule
The zero exponent rule states that any non-zero number raised to the power of zero equals 1. This might seem confusing at first, but here's why it works:

Let's think about the pattern when we decrease the exponent by 1 each time: \(3^3 = 27, \ 3^2 = 9, \ 3^1 = 3, \ 3^0 = 1\). Notice how each step involves dividing by the base (3 in this case). This pattern continues, and when we divide 3 by itself, we get 1. Therefore, \(3^0 = 1\). This rule holds for all non-zero numbers:
  • \(5^0 = 1\)
  • \(2^0 = 1\)
  • \(9^0 = 1\)), as shown in our exercise.

It's essential to remember this rule for solving various mathematical problems.
Powers of Numbers
Powers of numbers, also known as exponents, help us simplify expressions and solve problems efficiently. The key properties include:
  • Product of Powers: When multiplying two exponential numbers with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
  • Power of a Power: When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
  • Power of a Product: When raising a product to an exponent, apply the exponent to each factor: \( (ab)^m = a^m \times b^m\).
  • Quotient of Powers: When dividing two exponential numbers with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).

These rules make it easier to handle complex expressions involving powers. Remembering these properties will help you solve exponents-related problems in a more manageable and less error-prone way.

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Most popular questions from this chapter

Find each product. $$ \left(7 x+\frac{3}{7}\right)\left(7 x-\frac{3}{7}\right) $$

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ 7(4 m-3)(2 m+1) $$

Write each product as a sum of terms. Write answers with positive exponents only. Simplify each term. $$ \frac{1}{5 x}\left(5 x^{2}-10 x+45\right) $$

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

Evaluate. $$ 10,000(36.94) $$
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