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Chapter 4: Problem 11

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

Short Answer

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Step by step solution

01

Understand the Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero equals 1. This can be expressed mathematically as \(a^{0} = 1\) for any \(a eq 0\).
02

Apply the Zero Exponent Rule

Apply the zero exponent rule to the given expression. Here, the base is 9, which is a non-zero number, and the exponent is 0. Therefore, \(9^{0} = 1\).

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Key Concepts

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

Zero Exponent Rule
The Zero Exponent Rule is an essential part of understanding exponents in algebra. It states that any non-zero number raised to the power of zero equals 1. This is represented mathematically as: a^{0} = 1 \text{ for any } a eq 0This rule can be surprising at first, but it comes from consistent patterns observed in exponential expressions. For example, as you lower the exponent of a number like 2, you see that: \[ 2^{3} = 8 \]\[ 2^{2} = 4 \]\[ 2^{1} = 2 \]\[ 2^{0} = 1 \]Notice how each step essentially divides the previous result by the base, which in this case is 2.For any expression like 9^{0}, we immediately know the answer is 1, as long as the base is not zero.
Exponents in Algebra
Exponents in algebra simplify the representation and calculation of repeated multiplication. Instead of writing out multiple multiplications, you use an exponent to show how many times a base number is multiplied by itself. For example:\[ 2^{3} \]means 2 multiplied by itself three times: 2 * 2 * 2 = 8.It鈥檚 crucial to understand a few key exponent rules to work efficiently with them:
  • The Product Rule: \(a^{m} \cdot a^{n} = a^{m+n}\)
  • The Quotient Rule: \(\frac{a^{m}}{a^{n}} = a^{m-n}\)
  • The Power Rule: \((a^{m})^{n} = a^{m*n}\)
The zero exponent rule we examined is just one of several rules that make handling exponents in algebra easier and more systematic.
Mathematical Rules
Understanding mathematical rules helps you solve problems more effectively and avoid errors. Exponents are a key area where rules simplify complex calculations. Here are some fundamental rules beyond the zero exponent:
  • Negative Exponent Rule: \(a^{-n} = \frac{1}{a^{n}}\) (This makes sense if you think about reversing multiplication.)
  • One as an Exponent: Any number to the power of 1 equals itself: \(a^{1} = a\)
  • Exponent of One: 1 to any power is always 1, since multiplying 1 by itself any number of times never changes its value: \(1^{n} = 1\)
By using these rules, calculating and understanding expressions with exponents becomes more manageable. Knowing this, the zero exponent rule fits naturally as a part of your mathematical toolkit, ready for use whenever you encounter a problem involving exponents.

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Most popular questions from this chapter

During the \(2016-2017\) season, Broadway shows grossed a total of \(\$ 1.45 \times 10^{9}\). Total attendance for the season was \(1.33 \times 10^{7}\). What was the average ticket price (to the nearest cent) for a Broadway show? (Data from The Broadway League.)

\begin{aligned} &\text { Find the difference of the sum of } 5 x^{2}+2 x-3 \text { and } x^{2}-8 x+2 \text { and the sum of } 7 x^{2}-3 x+6\\\ &\text { and }-x^{2}+4 x-6 \end{aligned}

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Pollux, one of the brightest stars in the night sky, is 33.7 light-years from Earth. If one light-year is about \(6,000,000,000,000 \mathrm{mi}\) (that is, 6 trillion mi), about how many miles is Pollux from Earth? (Data from The World Almanac and Book of Facts.)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (m+6)^{2} $$
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