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Chapter 4: Problem 28

Evaluate. Assume that all variables represent nonzero real numbers. $$ -10^{0} $$

Short Answer

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Step by step solution

01

Understand the problem

Evaluate the given expression $$ \ -10^{0}.$$
02

Evaluate the exponent part

According to the laws of exponents, any number (except 0) raised to the power of 0 is equal to 1. Therefore, $$ 10^0 = 1.$$
03

Apply the negative sign

Now that the exponent part has been simplified to 1, apply the negative sign in front of the expression. $$ -10^0 = -1. $$

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Key Concepts

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

laws of exponents
To understand the laws of exponents, it's important to grasp how they work. Exponents are a way to express repeated multiplication. For instance, in \(10^3\), the number 10 is multiplied by itself three times (10 * 10 * 10). By learning the different laws of exponents, you can simplify complex expressions. Here are a few key laws:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \( (a^m)^n = a^{mn} \)
  • Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
  • Power of a Product: \( (ab)^n = a^n b^n \)
These laws make it easier to handle operations involving exponents. They are foundational for solving many mathematical problems.
negative sign application
Applying a negative sign in front of an expression can change its value significantly. In math, the negative sign indicates the opposite of a number. For example, while 5 is positive, -5 is its negative counterpart. This is crucial when evaluating exponents.
Consider the expression \( -10^0 \):
  • First, you evaluate the exponent: 10^0 = 1.
  • Next, apply the negative sign: - (1) = -1.
Understanding the proper placement of the negative sign can help avoid common mistakes. Always ensure to handle the exponent first before applying the negative sign if it's outside the exponentiation.
zero exponent rule
One of the fundamental rules in exponents is the zero exponent rule. According to this rule, any nonzero number raised to the power of zero equals 1. Mathematically, it can be expressed as:
\[ a^0 = 1 \] (for any \( a e 0 \))
This might seem counterintuitive at first, but it helps keep consistency within the laws of exponents. For example:
  • \(2^0 = 1\)
  • \(10^0 = 1\)
  • \((-5)^0 = 1\)
This rule simplifies calculations in many mathematical contexts. Always remember that the zero exponent rule only applies to nonzero numbers.

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Most popular questions from this chapter

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ \left(3 t-\frac{5}{4}\right)\left(3 t+\frac{5}{4}\right) $$

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{4}{5}\right)^{-2} $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ (0.2 x-1.4 y)^{2} $$

Perform the indicated operations. Subtract \(4 y^{2}-2 y+3\) from \(7 y^{2}-6 y+5\)

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{a^{-9} b^{6}}{a^{3} b^{-4}} $$
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