Negative Exponents
- Introduction
- What is a Negative Exponent?
- Negative Exponent Rules
- Understanding Negative Fraction Exponents
- Multiplying Powers with Negative Exponents
- Understanding Negative Exponents with Fractional Bases
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
When we take negative values for exponents, we call them negative exponents. This concept plays a crucial role in mathematics, particularly in areas like algebra and calculus. Negative exponents signify reciprocals of their corresponding positive exponents. Negative exponents are pivotal in simplifying expressions and equations involving exponents, making them fundamental in various mathematical contexts.
What is a Negative Exponent?
In mathematics, negative exponents are a fundamental concept that represents the reciprocal of their corresponding positive exponents. For example, \( 2^{-3} \) indicates that the reciprocal of `2`, that is \( \frac{1}{2} \), is to be divided by itself three times, resulting in \( \frac{1}{2^3} \), which simplifies to \( \frac{1}{8} \). Negative exponents play a significant role in simplifying expressions and solving equations. Understanding negative exponents is crucial for mastering various mathematical concepts.
Negative Exponent Rules
Negative Exponent Rule `1`:
When a number \( a \) has a negative exponent \( -n \), where \( n \) is a positive integer, the rule states to take the reciprocal of the base number \( a \) and multiply it according to the value of the exponent \( n \).
Example: Evaluate \( 3^{-2} \).
Solution:
Here, the base number is \( 3 \) and the exponent is \( -2 \).
Following the rule, \( 3^{-2} \) is rewritten as \( \frac{1}{3^2} = \frac{1}{3\times 3} = \frac{1}{9} \).
Therefore, the value of \( 3^{-2} \) is \( \frac{1}{9} \).
Negative Exponent Rule `2`:
When a number \( a \) appears in the denominator with a negative exponent \( -n \), where \( n \) is a positive integer, the result can be expressed as \( a \) multiplied by itself \( n \) times.
Example: Evaluate \( \frac{1}{5^{-4}} \).
Solution:
Here, the negative exponent is in the denominator of the fraction.
Following the rule, \( \frac{1}{5^{-4}} \) becomes \( 5^4 \), which is equal to \( 5 \times 5 \times 5 \times 5 = 625 \).
Hence, \( \frac{1}{5^{-4}} \) is equal to \( 625 \).
Understanding Negative Fraction Exponents
We may also come across negative fraction exponents, such as \(4^{-\frac{3}{2}}\). These too can be simplified into more manageable forms by employing exponent rules.
To begin, we can apply the rule \(a^{-n} = \frac{1}{a^n}\) to express \(4^{-\frac{3}{2}}\) as \(\frac{1}{4^{\frac{3}{2}}}\).
Next, we can further simplify by applying the exponent rule for fractional exponents. In this case, \(4^{\frac{3}{2}}\) can be rewritten as a radical \(\sqrt{4^3}\).
Continuing the simplification, we evaluate \(\sqrt{4^3}\), which equals \(\sqrt{64} = 8 \).
Finally, we have \(4^{-\frac{3}{2}} = \frac{1}{4^{\frac{3}{2}}} = \frac{1}{\sqrt{4^3}} = \frac{1}{8}\).
Multiplying Powers with Negative Exponents
When multiplying numbers with negative exponents, it's crucial to understand how the exponents interact and how to simplify the expression effectively. Here are an examples illustrating this process:
Example: Simplify \(3^{-2} \times 3^{-3}\)
Solution:
Begin by applying the product rule for exponents:
\(3^{-2} \times 3^{-3} = 3^{-2+(-3)} = 3^{-5}\)
Next, simplify \(3^{-5}\) by applying the negative exponent rule: \(3^{-5} = \frac{1}{3^5}\).
So, \(3^{-2} \times 3^{-3} = \frac{1}{3^5} = \frac{1}{243}\).
Understanding Negative Exponents with Fractional Bases
Sometimes, we do come across power terms with fractional bases and negative exponents.
For example : \( (\frac{2}{3})^{-5}\)
In order to simplify such term to a power term with positive exponent, we simply flip the base to make the negative exponent positive.
This means \( (\frac{2}{3})^{-5}\) is equivalent to \( (\frac{3}{2})^{5}\).
Example: Simplify \( (\frac{3}{4})^{-2}\).
Solution:
To simplify \( (\frac{3}{4})^{-2}\), we would first change the negative exponent to a positive exponent. To do so, we would flip the base \( \frac{3}{4}\) to \( \frac{4}{3}\) and change the exponent from `-2` to `2`.
Hence \( (\frac{3}{4})^{-2}\) `=` \( (\frac{4}{3})^{2}\).
\( (\frac{4}{3})^{2}\) `=` \( \frac{4^2}{3^2}\) `=` \( \frac{16}{9}\)
Hence \( (\frac{3}{4})^{-2}\) `=` \( \frac{16}{9}\).
Solved Examples
Example `1`. Simplify the expression \((5^3 + 2^4)^{-3}\).
Solution:
The given expression is \((5^3 + 2^4)^{-3}\).
\((5^3 + 2^4)^{-3} = (125 + 16)^{-3}\).
Simplify the expression inside the parentheses,
\((125 + 16)^{-3} = 141^{-3}\)
Apply the negative exponent's rule,
\( 141^{-3} = \frac{1}{141^3}\)
\( 141^3 = 2,357,261 \)
Thus, \((5^3 + 2^4)^{-3} = \frac{1}{2,357,261}\).
Example `2`. Find the value of \( x \) in `64/(2^((2-x))) = 16`.
Solution:
Given: `64/(2^((2-x))) = 16`
Rewrite \( 64 \) as \( 2^6 \) and \( 16 \) as \( 2^4 \) (since both are powers of `2`).
`(2^6)/(2^((2-x))) = 2^4`
Apply the quotient rule of exponents.
\( 2^{6-(2-x)} = 2^4 \)
Simplify the exponent.
\( 2^{6-2+x} = 2^4 \)
Combine like terms.
\( 2^{4+x} = 2^4 \)
If bases are the same, then exponents must be equal.
So, \( 4 + x = 4 \).
Solving this equation, we find \( x = 0 \).
Example `3`. Simplify the expression \( (\frac{3}{4})^{-2} + (6)^{-1} \) using negative exponent rules.
Solution:
By applying negative exponent rules, we can rewrite \( (\frac{3}{4})^{-2} \) as \( (\frac{4}{3})^{2} \) and \( (6)^{-1} \) as \( \frac{1}{6} \).
So, the given expression becomes:
\( (\frac{4}{3})^{2} + \frac{1}{6} \)
Now, simplify the square term:
\( (\frac{4}{3})^{2} = (\frac{16}{9}) \)
Thus, the expression simplifies to:
\( (\frac{16}{9}) + \frac{1}{6} \)
Now, find a common denominator for `9` and `6`:
\( (\frac{16}{9}) + (\frac{1}{6}) = (\frac{32}{18}) + (\frac{3}{18}) \)
Finally, add the fractions:
\( (\frac{32}{18}) + (\frac{3}{18}) = \frac{35}{18} \)
Therefore, \( (\frac{3}{4})^{-2} + (6)^{-1} \) simplifies to \( \frac{35}{18} \).
Example `4`. Simplify the expression \( (3^3) \times \frac{5^{-2}}{2^{-4}} \).
Solution:
Let’s first focus on \( \frac{5^{-2}}{2^{-4}} \).
Apply the rule for negative exponents to change the negative exponents into positive exponents.
\( \frac{5^{-2}}{2^{-4}} \) = \( \frac{2^4}{5^2} \)
So \( (3^3) \times \frac{5^{-2}}{2^{-4}} \) becomes \( (3^3) \times \frac{2^4}{5^2} \).
Simplifying the power terms, we get
\( (3^3) \times \frac{2^4}{5^2} = 27 \times \frac{16}{25} = \frac{432}{25} \)
Hence \( (3^3) \times \frac{5^{-2}}{2^{-4}} \) simplifies to \( \frac{432}{25} \).
Practice Problems
Q`1`. Simplify the expression \( (2^{-3} \times 3^{-2})^{-2} \).
- \( \frac{1}{72} \)
- \( \frac{1}{144} \)
- \( 5184 \)
- \( 72 \)
Answer: c
Q`2`. Evaluate \( \frac{1}{(4^{-2} \times 5^{-3})^{-1}} \).
- \( \frac{1250}{16} \)
- \( \frac{16}{1250} \)
- \( \frac{1}{2000} \)
- \( 625 \)
Answer: c
Q`3`. Simplify the expression \( (2^3 \times 3^{-2})^2 \).
- \( 36 \)
- \( 9 \)
- \( \frac{64}{36} \)
- \( \frac{64}{81} \)
Answer: d
Q`4`. Simplify \( \frac{5^{-2}}{6^{-3}} \).
- \( \frac{25}{216} \)
- \( \frac{216}{25} \)
- \( \frac{10}{18} \)
- \( \frac{18}{10} \)
Answer: b
Q`5`. Simplify \( (\frac{5}{3})^{-2} \times (\frac{6}{5})^{-4}\).
- \( \frac{49}{64} \)
- \( \frac{64}{49} \)
- \( \frac{432}{75} \)
- \( \frac{75}{432} \)
Answer: d
Frequently Asked Questions
Q`1`. What is a negative exponent?
Answer: A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, \(a^{-n}\) is equal to \(\frac{1}{a^n}\).
Q`2`. How do you simplify expressions with negative exponents?
Answer: To simplify expressions with negative exponents, apply the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\). Rewrite the expression with positive exponents and perform any necessary calculations.
Q`3`. Can negative exponents be in the denominator?
Answer: Yes, negative exponents can be in the denominator. When a term with a negative exponent is moved from the numerator to the denominator or vice versa, the sign of the exponent changes to positive.
Q`4`. What happens when you multiply numbers with negative exponents?
Answer: When multiplying numbers with negative exponents, add the exponents if the bases are the same. If the bases are different, simplify each term separately using the negative exponent rule, then multiply the results.
Q`5`. How do you solve equations involving negative exponents?
Answer: To solve equations involving negative exponents, isolate the term with the negative exponent and rewrite it with a positive exponent using the negative exponent rule. Then, proceed with solving the equation as usual, applying inverse operations to isolate the variable.