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Chapter 0: Problem 52

Evaluate each exponential expression. $$\frac{35 a^{14} b^{6}}{-7 a^{7} b^{3}}$$

Short Answer

Expert verified
\(-5a^{7}b^{3}\)

Step by step solution

01

Divide the coefficients

Divide 35 by -7. This will give -5 as the answer.
02

Subtract the exponent of \(a\)

Subtract the exponent of \(a\) in the denominator from the exponent of \(a\) in the numerator. In this case, this is \(a^{14 - 7}\) which simplifies to \(a^{7}\).
03

Subtract the exponent of \(b\)

Similarly subtract the exponent of \(b\) in the denominator from the exponent of \(b\) in the numerator. In this case, this is \(b^{6 - 3}\) which simplifies to \(b^{3}\).
04

Combine

Combine the results from step 1, 2 and 3. The simplified exponential expression is \(-5a^{7}b^{3}\).

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Key Concepts

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

Understanding Coefficients
Coefficients are the numbers placed in front of variables or variable expressions in a mathematical equation. They are crucial because they define how many times a term is to be taken. For instance, in the expression \( 2x \), the number 2 is the coefficient indicating that \( x \) is doubled or multiplied by 2.

In the problem \( \frac{35 a^{14} b^{6}}{-7 a^{7} b^{3}} \), the coefficients are 35 and -7. When dividing expressions, you'll start by focusing on these coefficients.

  • Divide 35 by -7.
  • Doing this calculation results in -5 as the new coefficient.
This forms the foundation for simplifying the entire expression by adjusting the numbers before manipulating the variables.
Subtraction of Exponents
Exponent subtraction is a key concept in algebra when dealing with divisions of like bases in exponential expressions. When you divide two expressions that have the same base, you subtract the exponents. This is a fundamental property that simplifies expressions significantly.

For instance, in the expression \( \frac{a^{14}}{a^{7}} \), you would subtract the exponent of the denominator from that of the numerator. So, \( 14 - 7 = 7 \), which simplifies to \( a^{7} \).

The same rule applies to other variables: in \( \frac{b^{6}}{b^{3}} \), you subtract 3 from 6 yielding \( b^{3} \).

  • This process of exponent subtraction helps in reducing or simplifying exponential expressions involving similar bases.
Always remember, the subtraction rule only applies if the bases (the letters) are the same.
Simplifying Expressions
In mathematics, simplifying expressions involves reducing them to their simplest form. This process cuts down complicated expressions into more manageable pieces without changing the value. After handling coefficients and exponents, you combine all simplified terms.

For the problem at hand, after dividing the coefficients and subtracting exponents:
  • The coefficient part of the expression was simplified to -5.
  • The exponents were reduced to \( a^{7} \) and \( b^{3} \).
By compiling these results, your simplified and final answer is \(-5a^{7}b^{3}\). This shows that simplifying expressions often involves multiple steps, each crucial for breaking down the expression into its simplest form.

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Most popular questions from this chapter

The formula for converting Celsius temperature, \(C,\) to Fahrenheit temperature, \(F\), is $$F=\frac{9}{5} C+32$$ If Fahrenheit temperature ranges from \(41^{\circ}\) to \(50^{\circ},\) inclusive, what is the range for Celsius temperature? Use interval notation to express this range.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{2}+\frac{2}{3}$$

This will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right)$$

What is the discriminant and what information does it provide about a quadratic equation?
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