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Chapter 14: Problem 33

Simplify. $$\frac{b^{-7}}{b^{12}}$$

Short Answer

Expert verified
The simplified form of the given expression \(\frac{b^{-7}}{b^{12}}\) is \(b^{-19}\).

Step by step solution

01

Identify the base and exponents

The given expression is \(\frac{b^{-7}}{b^{12}}\), where the base is \(b\), the exponent in the numerator is \(-7\), and the exponent in the denominator is \(12\).
02

Apply the exponent quotient rule

The quotient rule for exponents states: \(a^{m} 梅 a^{n} = a^{m-n}\), where \(a\) is the base, \(m\) and \(n\) are the exponents. Applying this rule to our expression, we get: \(\frac{b^{-7}}{b^{12}} = b^{-7-12}\)
03

Simplify the exponent by subtraction

Perform the subtraction in the exponent: \(-7-12=-19\) So, the simplified expression is: \(b^{-19}\) Thus, the simplified form of the given expression \(\frac{b^{-7}}{b^{12}}\) is \(b^{-19}\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions involves making an expression easier to understand or work with, by reducing it to its simplest form. Imagine that you are tidying up a space; you want things neat and organized鈥攊t鈥檚 the same with expressions. When it comes to exponents, simplifying can vastly make equations more manageable. A simplified expression will have fewer operations or a more straightforward representation. Some helpful practices when simplifying include:
  • Combining like terms
  • Using mathematical rules for exponents
For expressions with exponents, it鈥檚 crucial to understand the rules that govern them. These rules allow you to simplify with confidence, ensuring the expression maintains its integrity while looking less complex.
Negative Exponents
Negative exponents are less intimidating once you understand what they represent. A negative exponent indicates that the base is on the opposite side of a fraction: either moved from numerator to denominator or vice versa. Think of a negative sign in an exponent as a handy indicator telling you to "swap sides."
Let鈥檚 consider the example of negative exponents:
  • If you have a number with a negative exponent, like a^{-n}, you can rewrite it as \frac{1}{a^n}, essentially turning it into a fraction.
Thus,
  • \( b^{-7} \) becomes \( \frac{1}{b^7} \)
  • \( b^{-19} \) becomes \( \frac{1}{b^{19}} \)
This simplification helps visualize how the negative exponent affects the expression, making complex calculations easier to handle.
Quotient Rule for Exponents
The quotient rule for exponents is a powerful tool for simplifying expressions involving division of like bases. This rule states: if you divide two exponents by the same base, you can subtract the exponent in the denominator from the exponent in the numerator, written as \( a^m 梅 a^n = a^{m-n} \).
Here's how it applies to our example:
  • Starts with finding a common base, in this case, it鈥檚 \( b \).
  • In our expression \( \frac{b^{-7}}{b^{12}} \), apply the rule to get \( b^{-7-12} \).
  • The subtraction results in \( b^{-19} \), which is the simplified form.
Using this rule can turn complicated expressions into neat, manageable numbers by reducing them to a simpler exponent, making calculations quicker and easier for math learners. Always ensure the base is the same when applying this rule for it to work correctly.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log x=-4$$

Natsumi receives \(\$ 126\) per year in simple interest from three investments. Part is invested at \(2 \%\), part at \(3 \%\), and part at \(4 \% .\) There is \(\$ 500\) more invested at \(3 \%\) than at \(2 \% .\) The amount invested at \(4 \%\) is three times the amount invested at \(3 \% .\) Find the amount invested at each rate.

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Fill in the blank with the correct term. Some of the given choices will not be used. Descartes' rule of signs a vertical asymptote the leading-term test \(\quad\) an oblique the intermediate \(\quad\) asymptote value theorem direct variation the fundamental \(\quad\) inverse variation theorem of algebra a horizontal line a polynomial function a vertical line a rational function \(\quad\) parallel a one-to-one function \(\quad\) perpendicular a constant function a horizontal asymptote _____is a function that is a quotient of two polynomials.

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\ln x=-2$$
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