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Chapter 0: Problem 33

Simplify the expression and eliminate any negative exponent(s). $$ a^{9} a^{-5} $$

Short Answer

Expert verified
The simplified expression is \( a^4 \).

Step by step solution

01

Apply the Product of Powers Property

When multiplying two powers that have the same base, you can add the exponents. The expression \( a^9 a^{-5} \) can thus be simplified by adding the exponents: \( 9 + (-5) \).
02

Simplify the Expression

Add the exponents together: \( 9 + (-5) = 4 \). Hence, the expression simplifies to \( a^4 \).
03

Eliminate Negative Exponents

In this case, there are no negative exponents remaining once \( a^9 a^{-5} \) was simplified to \( a^4 \), so nothing further needs to be done.

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental concept in algebra that help us understand how many times a number, known as the base, is multiplied by itself. For example, in the expression \( a^9 \), the base \( a \) is multiplied by itself 9 times. Exponents are a useful shorthand for representing repeated multiplication, which helps simplify complex mathematical expressions.
  • The number at the top is called the exponent or power.
  • The number below it is known as the base.
  • An exponent of 1 means the base remains unchanged.
When dealing with exponents, it's important to remember that they follow specific properties which simplify the process of solving algebraic expressions.
Simplifying Expressions
Simplifying expressions involves rewriting them in a more concise form without changing their value. This process often makes it easier to work with the expressions. For algebraic expressions involving exponents, simplification can involve actions like combining like terms, especially when they have the same base.
One key rule when simplifying is to ensure that the steps taken do not change the meaning of the expression. This means you must follow the properties of exponents responsibly.
  • Combine powers with the same base by adding the exponents.
  • Look out for and eliminate negative exponents.
  • Ensure all terms are positive if the instruction demands it.
Simplifying expressions not only makes them shorter but also helps in performing mathematical operations more smoothly.
Product of Powers Property
The Product of Powers Property is a handy tool when working with exponents. This property states that when you multiply two powers with the same base, you can simply add their exponents. It helps streamline calculations by eliminating the need for multiplying each base separately.
The Product of Powers Property can be expressed as follows: if you have an expression like \( a^m \times a^n \), it equals \( a^{m+n} \).
  • This reflects the fact that you're multiplying the base \( a \) by itself \( m+n \) times in total.
  • Ensures consistency by adhering to basic algebraic rules.
  • Helps in reducing and solving equations efficiently.
In our original exercise, utilizing this property allowed a quick and error-free simplification of \( a^9 a^{-5} \) to \( a^4 \). Such efficiency is essential when solving more complex algebraic problems.

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Most popular questions from this chapter

\(89-96\) m State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ \frac{2}{4+x}=\frac{1}{2}+\frac{2}{x} $$

Factoring \(x^{4}+a x^{2}+b\) A trinomial of the form \(x^{4}+a x^{2}+b\) can sometimes be factored easily. For example \(x^{4}+3 x^{2}-4=\left(x^{2}+4\right)\left(x^{2}-1\right) .\) But \(x^{4}+3 x^{2}+4\) cannot be factored in this way. Instead, we can use the following method. \(x^{4}+3 x^{2}+4=\left(x^{4}+4 x^{2}+4\right)-x^{2} \quad\) Add and subtract \(x^{2}\) \(=\left(x^{2}+2\right)^{2}-x^{2} \quad\) Factor perfect square \(=\left[\left(x^{2}+2\right)-x\right]\left[\left(x^{2}+2\right)+x\right] \quad\) Difference of squares \(=\left(x^{2}-x+2\right)\left(x^{2}+x+2\right)\) Factor the following using whichever method is appropriate. (a) \(x^{4}+x^{2}-2\) (b) \(x^{4}+2 x^{2}+9\) (c) \(x^{4}+4 x^{2}+16\) (d) \(x^{4}+2 x^{2}+1\)

Write each number in decimal notation. $$ 3.19 \times 10^{5} $$

\(71-76\) m simplify the expression. (This type of expression arises in calculus when using the "quotient rule.") $$ \frac{(7-3 x)^{1 / 2}+\frac{3}{2} x(7-3 x)^{-1 / 2}}{7-3 x} $$

\(71-76\) m simplify the expression. (This type of expression arises in calculus when using the "quotient rule.") $$ \frac{\left(1-x^{2}\right)^{1 / 2}+x^{2}\left(1-x^{2}\right)^{-1 / 2}}{1-x^{2}} $$
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