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Chapter 0: Problem 51

Simplify each exponential expression. $$ \frac{25 a^{13} b^{4}}{-5 a^{2} b^{3}} $$

Short Answer

Expert verified
The simplified form of the given exponential expression is \(-5a^{11}b\).

Step by step solution

01

Separate Coefficients and Variables

Start by separating the coefficients (numbers) from the variables (letters with exponents): \( \frac{25}{-5} * \frac{a^{13}}{a^{2}} * \frac{b^{4}}{b^{3}} \)
02

Simplify Coefficients and Exponents

Then, simplify the separated parts: \(-5 * a^{13-2} * b^{4-3}\)
03

Simplify Further

Further simplification gives: \(-5 * a^{11} * b\)
04

Combine to Final Form

Finally, combine the separated parts into the final simplified form: \(-5a^{11}b\)

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Key Concepts

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

Exponential Expression
The term 'exponential expression' refers to a mathematical notation that conveys repeated multiplication of a base number raised to an exponent. In the context of our problem, the expression \(25 a^{13} b^{4}\) features both a constant (25) and variables (\(a\) and \(b\)) raised to exponents (13 and 4, respectively).

To simplify an exponential expression, it's crucial to understand that an exponent indicates how many times a base is multiplied by itself. For example, \(a^{13}\) means \(a\) is multiplied by itself 13 times.
Laws of Exponents
When dealing with exponential expressions, 'laws of exponents' are critical rules for carrying out algebraic simplification. They allow us to manipulate exponents in various ways to simplify expressions.
  • For division, \(a^m/a^n = a^{m-n}\) simplifies to the base raised to the difference of the exponents.
  • When multiplying like bases, you add exponents: \(a^m * a^n = a^{m+n}\).
  • The power of a product rule \( (ab)^n = a^n * b^n\) expresses that each base in a product is raised to the exponent.
Using these rules, we can tackle complex expressions systematically, such as the given problem, converting it into a more straightforward form.
Algebraic Simplification
Algebraic simplification involves combining like terms and reducing expressions to their simplest forms. This process frequently involves using the laws of exponents. For instance, in our exercise, we reduced \(a^{13}/a^{2}\) to \(a^{11}\) by subtracting the exponent of the denominator from the exponent of the numerator, as per the laws of exponents.

The goal is to minimize the complexity of the expression while keeping its value unchanged. This clarity can often reveal deeper insights into the problem and lead to easier computation or further manipulation.
Coefficient Simplification
Simplifying coefficients refers to reducing the numerical part of an algebraic expression to its simplest form. In our example, the coefficient simplification is accomplished by dividing the numerical coefficient of the numerator (25) by that of the denominator (-5).

This calculation results in \(25/-5 = -5\), which is then combined with the variables that have been simplified using their respective exponents. Coefficient simplification often precedes or is done simultaneously with the simplification of the variables' exponents to achieve a fully simplified result.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(8 x^{-6} y^{3}\right)^{\frac{1}{3}}\left(x^{\frac{5}{6}} y^{-\frac{1}{3}}\right)^{6} $$

Factor Completely. $$7 x^{4}+34 x^{2}-5$$

Factor completely. $$x^{2 n}+6 x^{n}+8$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ 4^{-2}<4^{-3} $$

Factor and simplify each algebraic expression. $$12 x^{-\frac{3}{4}}+6 x^{\frac{1}{4}}$$
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