Problem 38
Question
Simplify each exponential expression. $$ \frac{x^{30}}{x^{-10}} $$
Step-by-Step Solution
Verified Answer
The simplified form of the expression \(x^{30}/x^{-10}\) is \(x^{40}\).
1Step 1: Identify the given expression
The expression given in the problem is \(x^{30}/x^{-10}\). Here, \(x\) is the base and the powers are 30 and -10 respectively.
2Step 2: Apply the rule for dividing exponents
The rule for dividing exponential terms with the same base states that \(x^{m}/x^{n} = x^{m-n}\).Applying this rule to the given expression, we get \(x^{30-(-10)}\).
3Step 3: Simplify the expression
Subtracting a negative number is the same as adding that number. So, in our case, the expression simplifies to \(x^{30+10} = x^{40}\).
Key Concepts
Exponential ExpressionRules of ExponentsAlgebraic Simplification
Exponential Expression
An exponential expression is a mathematical representation that describes repeated multiplication of the same factor. It’s written as the base number raised to an exponent. For example, in the expression \(x^{n}\), \(x\) is the base and \(n\) is the exponent, indicating that \(x\) is multiplied by itself \(n\) times. Understanding the nature of exponential expressions is essential in algebra since they are prominently featured in equations, growth and decay problems, as well as in the study of functions.
In the exercise, \(\frac{x^{30}}{x^{-10}}\), \(x\) is the base, and the terms in the numerator and the denominator each have exponents. The overarching goal when working with exponential expressions is to simplify them, which can often reveal more information about the relationships between the variables involved. Simplifying exponential expressions involves applying the rules of exponents correctly to make the expression easier to work with or to interpret.
In the exercise, \(\frac{x^{30}}{x^{-10}}\), \(x\) is the base, and the terms in the numerator and the denominator each have exponents. The overarching goal when working with exponential expressions is to simplify them, which can often reveal more information about the relationships between the variables involved. Simplifying exponential expressions involves applying the rules of exponents correctly to make the expression easier to work with or to interpret.
Rules of Exponents
Basic Rules of Exponents
When it comes to simplifying exponential expressions, you need to apply the basic rules of exponents. These rules help in re-writing expressions in a more manageable way and are crucial for algebraic simplification.The pertinent rules of exponents used in the given exercise are:
- Product of Powers: To multiply two exponents with the same base, you add the exponents (\(x^{m} \cdot x^{n} = x^{m+n}\)).
- Quotient of Powers: To divide one exponent by another with the same base, you subtract the exponent in the denominator from the exponent in the numerator (\(x^{m}/x^{n} = x^{m-n}\)). This rule is directly applied in the exercise.
- Negative Exponent: A negative exponent means that the base is on the wrong side of a fraction, and to simplify, you take the reciprocal of the base (\(x^{-n} = 1/x^{n}\)).
- Power to a Power: When you have an exponent raised to another exponent, you multiply the exponents (\((x^{m})^{n} = x^{m \cdot n}\)).
- Zero Exponent: Any nonzero base with an exponent of zero equals one (\(x^{0} = 1\)).
Algebraic Simplification
Algebraic simplification is the process of reducing expressions to their simplest form. This does not just make the expression easier to understand or work with, but it can also reveal solutions to equations or simplify computations.
In our exercise, the steps taken to simplify \(\frac{x^{30}}{x^{-10}}\) showcase algebraic simplification at work. By applying the quotient rule (subtract the exponent in the denominator from the one in the numerator), we combine the two exponents into a single exponential expression. Here’s a tip for students to remember: when you encounter a negative exponent, think of subtraction as adding the opposite. So, for \(x^{30}/x^{-10}\), we effectively have \(x^{30+10}\), which is much easier to grasp.
Always remember that simplification is just a way to rewrite the same mathematical truth in a neater package, and the consistency of algebra ensures that the 'simplified' expression retains the same value as the original one. Keeping track of simplification steps and understanding the logic behind them, rather than just memorizing formulas, will make algebra less daunting and more logical for students.
In our exercise, the steps taken to simplify \(\frac{x^{30}}{x^{-10}}\) showcase algebraic simplification at work. By applying the quotient rule (subtract the exponent in the denominator from the one in the numerator), we combine the two exponents into a single exponential expression. Here’s a tip for students to remember: when you encounter a negative exponent, think of subtraction as adding the opposite. So, for \(x^{30}/x^{-10}\), we effectively have \(x^{30+10}\), which is much easier to grasp.
Always remember that simplification is just a way to rewrite the same mathematical truth in a neater package, and the consistency of algebra ensures that the 'simplified' expression retains the same value as the original one. Keeping track of simplification steps and understanding the logic behind them, rather than just memorizing formulas, will make algebra less daunting and more logical for students.
Other exercises in this chapter
Problem 38
Add or subtract terms whenever possible. $$ \sqrt{20}+6 \sqrt{5} $$
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In Exercises 15–58, find each product. $$ \left(3 x^{2}+4 x\right)\left(3 x^{2}-4 x\right) $$
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List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\
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add or subtract as indicated. $$ \frac{x^{2}+3 x}{x^{2}+x-12}-\frac{x^{2}-12}{x^{2}+x-12} $$
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