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Chapter 5: Problem 2

Write each expression with positive exponents only. Then simplify, if possible. $$9^{-2}$$

Short Answer

Expert verified
The expression \(9^{-2}\) expressed with positive exponents and simplified is \(1/81\)

Step by step solution

01

Understand the Power Rule for Negative Exponents

Per the rules of exponents, \(a^{-n} = 1/a^n\). This means we can rewrite \(9^{-2}\) as \(1/9^2\).
02

Apply the rule and express the number with positive exponents

By applying the rule from step 1, we can express \(9^{-2}\) as \(1/9^2\).
03

Simplify the expression

We can simplify the expression further by calculating the square of 9, which is 81. So, the expression \(1/9^2\) further simplifies to \(1/81\).

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Key Concepts

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

Negative Exponents
When you encounter negative exponents, it can seem a bit tricky, but they're not so bad once you understand the concept. Essentially, a negative exponent indicates that you need to take the reciprocal of the base to make the exponent positive.
So, if you have something like \(a^{-n}\), you can convert it to \(\frac{1}{a^n}\). This means you're just flipping the base to the other side of a fraction.
  • For example, \(9^{-2}\) becomes \(\frac{1}{9^2}\).
  • Remember: The base stays the same; the sign of the exponent just tells you if you'll write it as a fraction or not.
This approach helps simplify expressions and solve problems more easily, setting a solid foundation for working with exponents in different mathematical contexts.
Power Rule
The power rule is a key principle in dealing with exponents and is crucial for simplifying expressions. It's especially useful for converting expressions with negative exponents to a more standard form.
The rule is pretty straightforward: \(a^{-n} = \frac{1}{a^n}\). Here鈥檚 how it works:
  • Start with the base \(a\) raised to a negative exponent \(-n\).
  • Move the base \(a\) to the denominator and change the exponent to positive.
  • Now you have a simpler fraction where the exponent is positive.
For instance, \(9^{-2}\) becomes \(\frac{1}{9^2}\), which means the number 9 is squared and then reciprocated. Understanding this rule lays the groundwork for tackling more complex problems involving powers and roots.
Simplifying Expressions
Simplifying expressions is the final step in resolving problems with exponents. Once you've rewritten any negative exponents as positive, it's time to simplify the expression as much as possible. This generally involves performing any arithmetic under the exponent, then simplifying fractions if needed.
For the expression \(\frac{1}{9^2}\), you would:
  • First, calculate \(9^2\), which is \(81\).
  • This results in \(\frac{1}{81}\).
Boom! Your expression is now simplified. Doing this ensures that your calculations are tidy and easier to understand. Simplifying expressions is a key skill in algebra that helps in solving larger and more complicated equations.

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Most popular questions from this chapter

In Exercises \(79-82,\) simplify each expression. $$\frac{6 x^{3}(3 x-1)+5 x^{2}(6 x-3)}{3 x^{2}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$4^{-2}<4^{-3}$$

Perform the indicated operations. $$(y+1)\left(y^{2}-y+1\right)-(y-1)\left(y^{2}+y+1\right)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

Perform the indicated operations. $$4 x^{2}\left(5 x^{3}+3 x-2\right)-5 x^{3}\left(x^{2}-6\right)$$
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