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Chapter 4: Problem 38

Evaluate. Assume that all variables represent nonzero real numbers. $$ -5^{0}+5^{0} $$

Short Answer

Expert verified
Evaluate \(5^{0}\) which is 1, and then substitute back into expression \(-1 + 1 = 0\).

Step by step solution

01

Evaluate the Exponent

Recall that any number raised to the power of zero is 1. Thus: : \( 5^{0} = 1\) So both terms of the expression \(-5^{0}\) and 5^{0) are equal to one.
02

Apply Subtraction

Now substitute the evaluated exponent back into the expression: \(-1+1\).
03

Compute the Result

Compute the addition of the numbers: \(-1 + 1 = 0\).

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Key Concepts

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

zero exponent rule
Understanding the zero exponent rule is key when dealing with exponents. According to this rule, any nonzero number raised to the power of zero equals 1. This may seem surprising at first, but it holds true for all nonzero numbers.

For example:
  • \(5^{0} = 1\)
  • \((-3)^{0} = 1\)
  • \(101^{0} = 1\)
It does not matter which number you start with, as long as it is not zero. This rule simplifies many expressions quickly and is very useful in mathematics.

In the given exercise, when we see \(5^0\) and \(-5^0\), we immediately know both terms simplify to 1:
\[-5^0 + 5^0 = -1 + 1\]
simplifying expressions
Simplifying expressions involves reducing them to their most basic form. This often includes using rules of exponents, like the zero exponent rule. Let's break down the given expression:

The expression \(-5^0 + 5^0\) initially looks complicated, but with the zero exponent rule, it becomes easier to manage:

  • First, apply the zero exponent rule: \(-5^0 = -1\) and \(5^0 = 1\)
  • Next, substitute these values back into the expression: \(-1 + 1\)

This approach drastically simplifies what may first appear complex. Taking one step at a time, and knowing when to apply specific rules, helps you break down and simplify any expression.
addition and subtraction of integers
Addition and subtraction of integers are foundational arithmetic operations. These steps are straightforward but essential for simplifying expressions.

When adding or subtracting integers, it helps to remember the rules for combining positive and negative numbers:
  • Adding two positive numbers results in a positive number (e.g., \(3 + 4 = 7\)).
  • Adding two negative numbers results in a negative number (e.g., \(-3 + (-4) = -7\)).
  • Adding a positive and a negative number depends on their absolute values. Subtract the smaller absolute value from the larger one. The result takes the sign of the number with the larger absolute value (e.g., \(-3 + 4 = 1\) and \(3 + (-4) = -1\)).


In the case of the given exercise, the steps are as follows:
  • Combine the integers: \(-1 + 1\)
  • Since the numbers are opposites, their sum is zero: \(0\)
This simplified approach ensures accurate calculations and helps in resolving more complex mathematical problems.

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Most popular questions from this chapter

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (g h)(-3) $$

Find each product. $$ (3 r+2 s)\left(r^{3}+2 r^{2} s-r s^{2}+2 s^{3}\right) $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ (0.1 y+2.1)(0.5 y-0.4) $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ \left(q-\frac{3}{4} r\right)^{2} $$

Perform the indicated operations. $$ 5 k-(5 k-[2 k-(4 k-8 k)])+11 k-(9 k-12 k) $$
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