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Chapter 4: Problem 55

Find each product or quotient. Express using exponents. $$c^{5} \div c^{2}$$

Short Answer

Expert verified
The expression simplifies to \(c^3\).

Step by step solution

01

Identify the Operation

Notice that the exercise involves division (indicated by the division symbol \(\div\)). The base in both parts of the expression is the same, \(c\). We will use the properties of exponents to simplify this expression.
02

Recall the Quotient Rule for Exponents

The quotient rule for exponents states that when you divide like bases, you subtract the exponents: \(a^m \div a^n = a^{m-n}\). Apply this rule to the given problem \(c^5 \div c^2\).
03

Apply the Quotient Rule

Subtract the exponent of the denominator from the exponent of the numerator: \(5 - 2 = 3\). So, \(c^5 \div c^2 = c^{5-2} = c^3\).
04

Express the Final Result Using Exponents

After applying the rule of exponents, the expression simplifies to \(c^3\). Hence, \(c^5 \div c^2\) expressed using exponents is \(c^3\).

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Key Concepts

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

Understanding the Quotient Rule
In mathematics, exponents are shorthand for repeated multiplication. When you divide two powers that have the same base, you can apply a handy shortcut known as the "quotient rule". This rule helps simplify expressions efficiently.

The quotient rule states: if you have a base "a" raised to the power of "m" divided by the same base "a" raised to the power of "n", you subtract the exponents:
  • \( a^m \div a^n = a^{m-n} \)
So, remember, it only works when the bases are identical. It's a wonderful tool for reducing complexity in division problems involving exponents. By remembering to subtract the exponent of the denominator from the exponent of the numerator, you can simplify expressions with ease.
Division of Exponents
Dividing exponents is quite straightforward when the bases are the same. When faced with a division problem involving exponents, the first step is to confirm that the bases are identical.
  • For instance, in the expression \( c^5 \div c^2 \), both bases are "c".
Once you verify that the bases match, you just need to subtract the exponent in the denominator from the exponent in the numerator. It's that simple!

This subtraction gives us the new power of the base "c" in our example. So, by performing this division, you simplify the original expression to a new, simpler form鈥攑erfect for complex calculations that involve multiple steps.
Simplifying Expressions with Exponents
Simplifying expressions in algebra often involves using the rules of exponents to combine like terms or reduce the complexity of an expression. When dealing with exponents, one powerful tool is the subtraction of exponents when dividing.

To simplify \( c^5 \div c^2 \), follow these steps:
  • Confirm that the base "c" is the same in both terms.
  • Apply the quotient rule: subtract the exponents (5 minus 2), resulting in a new exponent of 3 for the base "c".
So, \( c^5 \div c^2 \) simplifies neatly into \( c^3 \).

Using this method, you can take seemingly complex exponent expressions and break them down into manageable, simple terms. Understanding these rules not only makes solving these problems easier but boosts your confidence in tackling algebraic expressions.

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Most popular questions from this chapter

Find each product. $$7.2 \times 100$$

Ninety-six centimeters is what part of a meter? $$\mathbf{A} \frac{3}{5}$$ $$B \frac{4}{5}$$ $$\text { C } \frac{23}{25}$$ $$\mathbf{D} \frac{24}{25}$$

Find each quotient. $$48 \div(-8)$$

CHALLENGE Is it true that \(\frac{23}{53}=\frac{23}{53}\) or \(\frac{2}{5} ?\) Explain why or why not.

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{6 x^{3}}{4 x^{4} y}$$
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