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Chapter 10: Problem 253

In the following exercises, simplify. $$\frac{\left(x^{2}\right)^{4}}{x^{5}}$$

Short Answer

Expert verified
The simplified expression is \( x^3 \).

Step by step solution

01

Apply the Power of a Power Rule

Use the rule \( (a^m)^n = a^{mn} \) to simplify the numerator. Here, \( x^2 \) is raised to the power of 4: \( (x^2)^4 = x^{2 \times 4} = x^8 \).
02

Rewrite the Expression

After simplifying the numerator, the expression becomes \( \frac{x^8}{x^5} \).
03

Apply the Quotient Rule

Use the rule \( \frac{a^m}{a^n} = a^{m-n} \) to simplify: \( \frac{x^8}{x^5} = x^{8-5} = x^3 \).

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Key Concepts

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

Power of a Power Rule
When simplifying expressions with exponents, the **Power of a Power Rule** is a helpful tool. This rule is used when we have a base raised to an exponent, which is then raised to another exponent. Mathematically, it is expressed as \((a^m)^n = a^{mn}\).
Quotient Rule
The **Quotient Rule** simplifies the division of two powers with the same base. According to this rule, \(\frac{a^m}{a^n} = a^{m-n}\). Basically, you subtract the exponent of the denominator from the exponent of the numerator.
Exponentiation
Exponentiation is a way to denote repeated multiplication of a number by itself. The general form is \(a^n\), where \(a\) is the base and \(n\) is the exponent. The rules above, such as the Power of a Power Rule and the Quotient Rule, help us simplify expressions involving exponentiation.

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Most popular questions from this chapter

Factor the greatest common factor from each polynomial. $$-6 x^{2}-30 x$$

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