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Chapter 6: Problem 381

Simplify. $$\frac{\left(x^{4}\right)^{5}}{x^{15}}$$

Short Answer

Expert verified
\( x^5 \)

Step by step solution

01

- Apply the Power Rule

Start by applying the power rule \( (a^m)^n = a^{mn} \) to the numerator \( (x^4)^5 \). This results in \[ (x^4)^5 = x^{4 \times 5} = x^{20} \]
02

- Combine the Expression

Replace the numerator with the simplified result from Step 1, so the expression becomes \[ \frac{x^{20}}{x^{15}} \]
03

- Apply the Quotient Rule

Use the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \) to simplify the expression. Subtract the exponent in the denominator from the exponent in the numerator: \[ \frac{x^{20}}{x^{15}} = x^{20-15} = x^5 \]

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Key Concepts

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

Power Rule Explanation
The Power Rule in exponents is an important tool for simplifying expressions. It states that when you raise a power to another power, you multiply the exponents. In mathematical form, it can be written as: \[ (a^m)^n = a^{m \times n} \].
Let's break it down with our example: \[ (x^4)^5 \]. Using the Power Rule, we multiply the exponents 4 and 5 together: \[ x^{4 \times 5} = x^{20} \]. This step allows you to simplify expressions where an exponent is raised to another exponent.
Quotient Rule and Its Application
The Quotient Rule helps simplify expressions involving division of the same base with different exponents. The rule states \[ \frac{a^m}{a^n} = a^{m - n} \]. By subtracting the exponent in the denominator from the exponent in the numerator, you simplify the expression to a single exponent.
In our problem, we have \[ \frac{x^{20}}{x^{15}} \]. Applying the Quotient Rule, we get: \[ \frac{x^{20}}{x^{15}} = x^{20-15} = x^5 \]. This simply shows that the remaining exponent after the division is the difference of the two original exponents.
Simplifying Exponents in Practice
Simplifying exponents can make complex expressions much easier to work with. Start by using the Power Rule and Quotient Rule, which we've covered. Combining these rules helps to reduce expressions to their simplest form without losing any information.
Here's a quick review:
  • Apply the Power Rule to any base with an exponent raised to another exponent.
  • Simplify the expression first, before moving on to division or other operations.
  • Use the Quotient Rule to deal with division by subtracting exponents when the bases are the same.
In our initial exercise, we used these steps to go from \[ \frac{\big(x^{4}\big)^{5}}{x^{15}} \] to just \[ x^5 \]. This clearly shows how powerful these rules are in breaking down and simplifying complex exponent expressions.

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Most popular questions from this chapter

Simplify. (a) \(\left(\frac{7}{2}\right)^{-3}\) (b) \(\left(-\frac{3}{x y^{2}}\right)^{-3}\)

Divide each polynomial by the monomial. $$\frac{72 r^{5} s^{2}+132 r^{4} s^{3}-96 r^{3} s^{5}}{12 r^{2} s^{2}}$$

Divide each polynomial by the binomial. $$\left(x^{3}-216\right) \div(x-6)$$

Divide each polynomial by the binomial. $$\left(x^{2}+2 x-30\right) \div(x-5)$$

Mixed Practice (a) \(p^{4} \cdot p^{6}\) (b) \(\left(p^{4}\right)^{6}\)
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