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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 expression is \(-5 a^{11} b\).

Step by step solution

01

Simplify Coefficients

Calculate the result of \(\frac{25}{-5}\). The result will be the coefficient of the simplified expression.
02

Simplify Terms with Base \(a\)

Apply the exponential laws on the terms \(a^{13}\) and \(a^{2}\). The law states that \((a^{m})/(a^{n})=a^{m-n}\), subtract the exponent of the denominator from the exponent of the numerator. You should get \(a^{11}\).
03

Simplify Terms with Base \(b\)

Lastly, apply the exponential laws on the terms \(b^{4}\) and \(b^{3}\). The law states that \((b^{m})/(b^{n})=b^{m-n}\), subtract the exponent of the denominator from the exponent of the numerator. You should get \(b^{1}\) or simply \(b\).

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Key Concepts

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

Simplifying Coefficients
When simplifying exponential expressions, the first step is often to simplify the coefficients. Coefficients are the numerical parts of the terms that are not attached to any variable or exponent. In the expression \( \frac{25 a^{13} b^{4}}{-5 a^{2} b^{3}} \), the coefficients are 25 and -5. To simplify them, you perform the division \( \frac{25}{-5} \) which results in -5.

This means the simplified expression will start with a coefficient of -5 in the numerator. Simplifying coefficients is a crucial initial step, as it sets the stage for simplifying the rest of the expression that involves variables and their exponents.
Exponential Laws
Exponential laws or rules are fundamental principles used to manipulate and simplify expressions with exponents. One key rule to remember is:
  • If you have the same base and you鈥檙e dividing, subtract the exponents. For example,
    • \( \frac{a^{m}}{a^{n}} = a^{m-n} \)
  • This rule applies to both numeric and variable base terms.
In our expression, after simplifying the coefficients, we have \( a^{13} \) divided by \( a^{2} \). Using the exponential law, you subtract the exponent of 2 from 13, giving you \( a^{11} \).

Applying exponential laws is vital in maintaining the integrity of the expression while making it simpler and more compact.
Base Terms Simplification
Base terms simplification involves making the variables in an expression easier to work with by applying the exponential laws. After dealing with coefficients and starting with the expression \( -5a^{11}b^{4} \), we now focus on the variable \( b \). Apply the rule \( \frac{b^{4}}{b^{3}} = b^{4-3} = b^{1} \), which can simply be written as \( b \).

The simplified form of the entire expression, after applying the base terms simplification, is \( -5a^{11}b \). Focusing on each base separately helps in understanding and methodically simplifying rather than approaching the problem in a complex lump sum.

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Most popular questions from this chapter

Find all numbers that must be excluded from the domain of each rational expression. $$ \frac{x+5}{x^{2}-25} $$

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state the name of the property illustrated. $$ (x+4)+[-(x+4)]=0 $$

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{x^{2}-8 x+16}{3 x-12} $$

Use the quotient rule to simplify the expressions in Exercises \(17-26 .\) Assume that \(x>0\) $$\frac{\sqrt{150 x^{4}}}{\sqrt{3 x}}$$
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