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Chapter 5: Problem 15

Express using positive exponents and, if possible, simplify. $$a^{-3}$$

Short Answer

Expert verified
\(a^{-3} = \frac{1}{a^3}\)

Step by step solution

01

Understand Negative Exponents

Recall that a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. Specifically, for any non-zero number a and a positive integer n, \(a^{-n} = \frac{1}{a^n}\).
02

Apply the Negative Exponent Rule

Apply the rule to the given expression. Here, the expression is \(a^{-3}\), which means we need to take the reciprocal of \(a\) and raise it to the power of 3. This results in \(a^{-3} = \frac{1}{a^3}\).
03

Simplify the Expression

Since there are no common factors to simplify further, \(\frac{1}{a^3}\) is already in its simplest form.

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Key Concepts

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

Reciprocal
When you see a negative exponent, it's important to think about reciprocals. A reciprocal is what you get when you flip a number. For example, the reciprocal of 2 is \(\frac{1}{2}\), and the reciprocal of 5 is \(\frac{1}{5}\). When a number has a negative exponent, like \(a^{-3}\), you take the reciprocal of the base and then apply the positive exponent. That鈥檚 why \(a^{-3}\) turns into \(\frac{1}{a^3}\). Just flip it inside out and make the exponent positive! Keep practicing reciprocals with different numbers. It will make negative exponents much easier to handle.
Simplifying Expressions
Simplifying expressions is all about making them easier to understand and work with. After converting \(a^{-3} \) into \(\frac{1}{a^3}\), look to see if you can simplify further. Sometimes you'll have to combine like terms or reduce fractions. In this case, \(\frac{1}{a^3}\) is already in simplest form. Simplifying isn鈥檛 always about finding a simpler expression, but also making it more useful and clear. Keep an eye out for opportunities to make your work clean and concise. This skill is essential for tackling more complex problems down the road.
Positive Exponents
Positive exponents represent how many times to multiply the base by itself. For example, with \(a^3\), you multiply \(a\) by itself three times: \(a \times a \times a\). Converting a negative exponent like \(a^{-3}\) into \( \frac{1}{a^3}\) means you're now working with a positive exponent in the denominator. Using positive exponents makes calculations more straightforward and avoids confusion. It鈥檚 always clearer to express results with positive exponents whenever possible. This approach will help you follow mathematical conventions and make your answers universally understandable.

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Most popular questions from this chapter

Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D6 contains the formula $$ =\mathrm{Al}-0.2 * \mathrm{B} 1+0.3^{*} \mathrm{Cl} $$ What is the value in \(\mathrm{D} 6\) if the value in \(\mathrm{Al}\) is \(10,\) the value in \(\mathrm{B} 1\) is \(-3,\) and the value in \(\mathrm{Cl}\) is \(30 ?\)

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{x^{2} y}{z^{3}}\right)^{4} $$

What is the degree of \(\left(5 m^{5}\right)^{2} ?\)

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{5}{3-x}\\\ &g(x)=\frac{x}{4 x-1} \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=5 x-1\\\ &g(x)=2 x^{2} \end{aligned} $$
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