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Chapter 1: Problem 68

Write an equivalent expression with negative exponents. $$ \frac{1}{(5 x)^{5}} $$

Short Answer

Expert verified
(5x)^{-5}

Step by step solution

01

Rewrite the Exponential Expression

To express a fraction with a positive exponent in the denominator using negative exponents, take the reciprocal of the expression and change the sign of the exponent. Given the expression is \ \(\frac{1}{(5x)^5}\ \), write it as \ \((5x)^{-5}\) \.

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Key Concepts

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

Equivalent Expressions
An equivalent expression is a different way of writing the same mathematical concept. Even though the expressions look different, they hold the same value. For instance, in this exercise, we were given the fraction \(\frac{1}{(5x)^5}\). By applying the rules of negative exponents, we converted it to \( (5x)^{-5} \). Both expressions are equivalent as they represent the same number. This is a useful technique in simplifying expressions and solving equations. It helps to understand that math has multiple ways to represent the same idea.
  • Start with \(\frac{1}{(5x)^5}\)
  • Take the reciprocal of \( (5x)^5 \) giving us \( (5x)^{-5} \)
This process provides valuable flexibility in manipulating and understanding mathematical expressions.
Exponential Expressions
Exponential expressions involve numbers or variables raised to a power. The power, or exponent, shows how many times to multiply the base by itself. In \( (5x)^5 \), \( 5x \) is the base and \( 5 \) is the exponent. The negative exponent rule simplifies expressions and equations by providing alternate forms. Instead of \(\frac{1}{(5x)^5}\), we express it as \( (5x)^{-5} \). This method can make calculations easier and more intuitive.
  • Base: The number or variable being multiplied
  • Exponent: Indicates how many times to multiply the base by itself
  • Negative exponent: Represents the reciprocal of the base and a positive exponent
A strong understanding of these rules is essential for higher-level math concepts and problems.
Reciprocal
A reciprocal is the inverse of a number or expression. If you have a number \(a\), its reciprocal is \( \frac{1}{a} \). For example, the reciprocal of \( (5x)^5 \) is \(\frac{1}{(5x)^5}\). This concept is key when dealing with negative exponents. A negative exponent indicates taking the reciprocal of the base raised to the opposite positive exponent. Hence, \((5x)^{-5}\) is the reciprocal of \((5x)^5\). Understanding reciprocals helps simplify and solve equations more efficiently.
  • Positive exponent: Just multiply the base
  • Negative exponent: Take the reciprocal of the base with a positive exponent
  • Reciprocal of \(a\): \( \frac{1}{a} \)
Mastering reciprocals expands your ability to handle various mathematical expressions and equations.

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Most popular questions from this chapter

Solve and check. The symbol \(\sqsubseteq\) indicates an exercise designed to be solved with a calculator. $$ 6 x-\\{5 x-[7 x-(4 x-(3 x+1))]\\}=6 x+5 $$

Write an equivalent expression using an associative law. $$ (3 x) y $$

Divide. $$ \left(-\frac{2}{7}\right) \div(-1) $$

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ 196 \in \mathbb{N} $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ -(-3)^{2} $$
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