Podcast
Questions and Answers
What is the value of $5^{-2}$?
What is the value of $5^{-2}$?
The expression $x^{-3}$ can be simplified to $1/x^{3}$.
The expression $x^{-3}$ can be simplified to $1/x^{3}$.
True
What does a negative exponent indicate?
What does a negative exponent indicate?
It indicates the reciprocal of the base raised to the positive exponent.
The expression $a^{-n}$ can be rewritten as ________.
The expression $a^{-n}$ can be rewritten as ________.
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Match the following expressions with their equivalent values:
Match the following expressions with their equivalent values:
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Study Notes
Reverse Percentage
- Reverse percentage problems involve finding the original value after a percentage change has been applied.
- Finding the original value is often referred to as "finding the 100% value".
- To solve these problems, work backward from the given percentage and its effect on the initial value.
- Example: If a price is reduced by 20% and the new price is $80, the original price can be found by dividing the new price by 80% (which is 0.8).
- Formula: Original value = New value / (1 - percentage decrease)
- In the example, Original price = $80 / 0.8 = $100
- Note that the opposite is also true
- If a quantity is increased by X% the new value is (1 + X/100) * the original value
- Example: an object is increased by 20%, so the new value is 1.2 * the original value
Rule of Negative Indices
- Negative indices represent reciprocals.
- If a number has a negative exponent, it means to take the reciprocal of that number raised to the positive exponent.
- Formula: a-n = 1/an
- Important note: The base of the exponent is significant and affects the entire expression.
- Example: 5-2 = 1/52 = 1/25
- Further Explanation: This rule is fundamental to simplifying and evaluating expressions involving negative exponents. It signifies a relationship between division and powering.
- Example Application: To solve an expression like (1/3)-2, you raise the reciprocal (3) to the positive power (2) for 32 = 9.
- Another example: when dealing with x-n/ y-m, rewrite it as ym/ xn; this is useful for simplifying and collecting terms.
- Extension to more complex expressions: This principle extends to situations involving variables and multiple terms with negative exponents. Careful attention is needed to the placement of terms in the fraction and to address both positive and negative exponents.
- Practical use: Negative indices simplify calculations involving fractions, roots, scientific notation, and further mathematical computations.
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Description
This quiz explores the concepts of reverse percentage and the rule of negative indices. Understand how to find the original value after a percentage change and the application of negative exponents in mathematics. Test your knowledge with practical examples and calculations.
