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Chapter 5: Problem 18

Use the zero and negative exponent rules to simplify each expression. \(-9^{0}\)

Short Answer

Expert verified
The simplified expression is 1.

Step by step solution

01

Apply the zero exponent rule

The zero exponent rule states that any non-zero number to the power of zero is one. Therefore, the expression \(-9^{0}\) can be simplified as follows: \[ -9^{0} = 1 \]

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Key Concepts

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

Zero Exponent Rule
Did you know that any non-zero number raised to the power of zero is always one? This might seem surprising at first! Nonetheless, it鈥檚 a fundamental rule in mathematics. It means whenever you see something like
  • \(-9^0\)
  • \(5^0\)
  • even \((1000)^0\)
you can immediately know the result is just 1.

But why does this work in the first place? Think about how exponent rules continue from what happens when we decrease the power. For instance:
  • \(9^3 = 729\)
  • \(9^2 = 81\)
  • \(9^1 = 9\)
  • and then \(9^0 = 1\)
Each time we subtract one from the exponent, we divide the result by the base. Hence, knowing that the pattern of division continues helps us understand this rule! So whenever you apply the zero exponent rule, you are simply continuing that pattern which results in 1.
Negative Exponents
Negative exponents might seem a bit tricky at first, but once you understand the concept, it's straightforward to apply! When you see a negative exponent, it signifies the reciprocal of the base raised to the absolute value of the exponent.

For example, when you have a base like \(a\) with a negative exponent \(-n\), such as \(a^{-n}\), the expression becomes:
  • \(a^{-n} = \frac{1}{a^n}\)
This tells us we're flipping the base into the denominator. So, it's like the inverse operation of raising to the power!

Some quick examples:
  • \(10^{-1} = \frac{1}{10}\)
  • \(2^{-3} = \frac{1}{8}\)
  • and even \(5^{-2} = \frac{1}{25}\)
Mastering this helps simplify expressions with negative exponents effortlessly, converting them into positive ones for calculation.
Simplifying Expressions
Simplifying expressions means breaking down a math problem into its simplest form. It involves using different rules and techniques to make a complex expression easier to understand and solve.

Here are some ways to simplify expressions:
  • Use the zero exponent rule to simplify anything to 1 when the exponent is zero as shown above.
  • Apply the negative exponent rule to turn negative powers into fractions.For instance, transform \(4^{-2}\) into \(\frac{1}{4^2} = \frac{1}{16}\).
  • Combine like terms and arrange the expression in a simple way.
    • For instance, \(2x + 4x = 6x\).
  • Finally, make sure your answer is as reduced as possible, which means it cannot be simplified any further.
By practicing these techniques, you'll become more confident in simplifying many algebraic expressions with ease!

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Most popular questions from this chapter

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(0.0004,-0.004,0.04,-0.4, \ldots\)

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to \(2015 .\) Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,18.4 \%\) of American women ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.6\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American women ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American women ages 25 and older who will be college graduates by 2029 .

The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is given by$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right),$$in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by$$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r},$$in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(4,-12,36,-108, \ldots\)

What is the common difference in an arithmetic sequence?

Company A pays \(\$ 44,000\) yearly with raises of \(\$ 1600\) per year. Company B pays \(\$ 48,000\) yearly with raises of \(\$ 1000\) per year. Which company will pay more in year 10 ? How much more?
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