6 Questions
What is the result of converting the negative exponent in the equation $y^{-3} = 1/125$ to a positive exponent?
$y^3 = 125$
When simplifying the expression $5x^{-2}$, what is the correct simplified form?
$5/x^2$
In the expression $(a^2b^{-3})/(ab)$, what is the result after applying the quotient rule for negative exponents?
$a^{1}b^{-2}$
If you have to simplify $7x^{-3}y^4$, what would be the correct simplified form?
$7y/x$
What is the result of multiplying $6x^{-2} * 3x^{-4}$ and simplifying using the product rule for negative exponents?
$18x^{-6}$
If we have the equation $2y^{-1} = 16$, what is the value of $y$ after converting the negative exponent to a positive exponent?
$y = 8$
Study Notes
Working with Negative Integral Exponents in Mathematics
Negative integral exponents in mathematics can lead to counterintuitive results at first glance, but they play a crucial role in solving equations, simplifying expressions, and manipulating functions. Let's delve into these concepts and uncover their foundations.
Solving Equations Involving Negative Exponents
When solving equations with negative exponents, the key is to rewrite the equation in a form where the exponents are nonnegative. For example, if we have (x^{-2} = 4), we can rewrite this as (x^2 = 1/4), which we can solve traditionally.
Simplifying Expressions with Negative Exponents
Negative exponents signify reciprocals. For instance, (x^{-1} = \frac{1}{x}), (x^{-2} = \frac{1}{x^2}), and so on. This convention helps us simplify expressions involving negative exponents. For example, ({(2x)}^{-3} = \frac{1}{(2x)^3} = \frac{1}{8x^3}).
Applying the Quotient Rule with Negative Exponents
When dealing with quotients of expressions with negative exponents, we must apply the quotient rule, which states that ((a^m/a^n) = a^{m-n}). For example, (\frac{x^{-2}}{x^{-3}} = x^{2-3} = x^{-1}).
Applying the Product Rule with Negative Exponents
The product rule, which states that (a^m \cdot a^n = a^{m+n}), applies to expressions with negative exponents as well. For instance, (x^{-2} \cdot x^{-3} = x^{-2-3} = x^{-5}).
Converting Negative Exponents to Positive Exponents
Converting negative exponents to positive exponents can help make calculations easier, especially when dealing with fractions or ratios. For example, if we have (\frac{3}{x^{-2}} = 3 \cdot x^2). This conversion allows us to perform calculations more intuitively and efficiently.
Negative exponents, especially when combined with integral exponents, can sometimes seem confusing. However, once you grasp the basic principles and rules, you'll find that they open up a world of possibilities—from solving complex equations to simplifying complex expressions.
Explore the principles, rules, and applications of negative integral exponents in mathematics. Learn how to solve equations, simplify expressions, and apply rules like the quotient rule and product rule when dealing with negative exponents.
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