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Chapter 5: Problem 48

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(b^{4}\right)^{-3}\)

Short Answer

Expert verified
\(\frac{1}{b^{12}}\)

Step by step solution

01

Apply the Power Rule for Exponents

The expression is \(\left(b^{4}\right)^{-3}\). According to the power rule of exponents, \(\left(a^m\right)^n = a^{m \cdot n}\). Apply this rule to get \(b^{4 \cdot (-3)} = b^{-12}\).
02

Convert Negative Exponent to Positive Exponent

To rewrite \(b^{-12}\) without a negative exponent, use the property \(a^{-n} = \frac{1}{a^n}\). Therefore, \(b^{-12} = \frac{1}{b^{12}}\). This makes the exponent positive.

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Key Concepts

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

Power Rule
Understanding the power rule for exponents makes working with expressions much easier. The power rule states that when you raise a power to another power, you multiply the exponents together. This can be written as \( \left(a^m\right)^n = a^{m \cdot n} \).
In simpler terms, if you have a base raised to a power and that entire expression is then raised to another power, you need to multiply those two exponents to simplify. For the given expression, \( \left(b^{4}\right)^{-3} \), the power rule applies, meaning:
  • Raise \( b^4 \) to the \( -3 \) power. Multiply the exponents: \( 4 \times (-3) = -12 \).
      • This results in \( b^{-12} \).
      It's a straightforward process that helps reduce complex expressions into simpler ones.
Negative Exponents
Negative exponents can be a tricky concept, but they don't have to be confusing. The principle behind them is simple: any expression with a negative exponent represents . This means that \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \).
For instance, in our example, \( b^{-12} \) indicates \( \frac{1}{b^{12}} \).
  • This transformation from a negative exponent to a fraction is crucial in simplifying further and achieving the final expression in its base positive exponent form.
Getting comfortable with flipping negative exponents into fractions helps in numerous algebraic operations, ensuring expressions are always in a clean and usable form.
Simplifying Expressions
Simplifying expressions is a fundamental skill in mathematics. It involves transforming a complex expression into its simplest and most digestible form. For algebraic expressions, this often includes using rules and properties of exponents efficiently.
  • It involves steps like applying the power rule correctly to multiply exponents.
  • Changing negative exponents into fractions to avoid any negative powers.
  • Finally, ensuring that all parts of the expression are expressed using positive exponents.
By simplifying, complex expression \( \left(b^4\right)^{-3} \) morphs into \( \frac{1}{b^{12}} \).
This simpler form is easier to comprehend and use in further calculations or when plugging into larger equations. Practicing these techniques helps in achieving precision and eliminates any confusion in solving mathematical problems.

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Most popular questions from this chapter

(a) Because \(\frac{4}{5}=0.8\), we can evaluate \(10^{\frac{4}{5}}\) by evaluating \(10^{0.8}\), which involves a shorter sequence of "calculator steps." Evaluate parts (b), (c), (d), (e), and (f) of Problem 96 and take advantage of decimal exponents. (b) What problem is created when we try to evaluate \(7^{\frac{4}{3}}\) by changing the exponent to decimal form?

Use your calculator to evaluate each of the following. (a) \(\sqrt[3]{1728}\) (b) \(\sqrt[3]{5832}\) (c) \(\sqrt[4]{2401}\) (d) \(\sqrt[4]{65,536}\) (e) \(\sqrt[5]{161,051}\) (f) \(\sqrt[5]{6,436,343}\)

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ \frac{360,000,000}{0.0012} $$

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (3.14)\left(10^{10}\right) $$

For Problems \(1-18\), write each of the following in scientific notation. \(0.00000000003\)
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