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Chapter 6: Problem 392

Simplify. $$\left(\frac{w^{5}}{x^{3}}\right)^{8}$$

Short Answer

Expert verified
\( \frac{w^{40}}{x^{24}} \)

Step by step solution

01

Understand the power of a quotient property

When simplifying expressions of the form \(\frac{a}{b}\bigg)^n\), apply the power to both the numerator and the denominator. This means \(\bigg(\frac{w^5}{x^3}\bigg)^8\) becomes \( \frac{(w^5)^8}{(x^3)^8} \).
02

Apply the power rule for exponents

The power rule for exponents states that \((a^m)^n = a^{mn}\). Using this rule, raise each base to the power of 8: \((w^5)^8 = w^{5 \times 8} = w^{40}\) and \((x^3)^8 = x^{3 \times 8} = x^{24}\).
03

Rewrite simplified expression

Combine the previously calculated results to get the simplified form: \( \frac{w^{40}}{x^{24}} \).

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Key Concepts

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

Power of a Quotient Property
When you encounter an expression like \(\bigg(\frac{w^5}{x^3}\bigg)^8\), you're dealing with the power of a quotient property.
This property helps you simplify expressions by applying the exponent to both the numerator and the denominator.
Think of it like distributing the exponent. So for \(\bigg(\frac{w^5}{x^3}\bigg)^8\), you need to individually raise each term inside the fraction to the 8th power.

Your expression changes form step by step:
  • First, rewrite \(\bigg(\frac{w^5}{x^3}\bigg)^8\) as \(\frac{(w^5)^8}{(x^3)^8}\).
  • The numerator becomes \( (w^5)^8\).
  • The denominator becomes \( (x^3)^8\).

This makes it easier to see what comes next. You're simplifying in a way that breaks the problem into smaller, manageable parts.
Power Rule for Exponents
Once you've rewritten your expression using the power of a quotient property, it's time to apply the power rule for exponents.
The power rule tells us that \( (a^m)^n = a^{mn}\). In simpler terms, you multiply the exponents together.

Using this rule for our expression \(\frac{(w^5)^8}{(x^3)^8}\):
  • Apply the rule to the numerator: \( (w^5)^8 = w^{5 \times 8} = w^{40} \).
  • Apply the rule to the denominator: \( (x^3)^8 = x^{3 \times 8} = x^{24} \).

So, combining these results gives us a transformed expression \ ( \frac{ w^{40} }{ x^{24} }) \. Each term in the original fraction is now raised to the power applied appropriately based on the rule.
Simplifying Algebraic Expressions
Finally, let's put everything together to simplify the algebraic expression entirely.
At this point, you're looking at \( \frac{ w^{40} }{ x^{24} } \).
Simplifying algebraic expressions means making them as straightforward as possible while maintaining the same value.

In our example:
  • The numerator has been simplified to \( w^{40} \).
  • The denominator has been simplified to \( x^{24} \).

The expression \( \frac{ w^{40} }{ x^{24} } \) is now in its simplest form, as there are no further common factors or reductions possible.
This concise and clear form represents the final simplified version of the original problem, showing the power of working through each step methodically.

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Most popular questions from this chapter

Divide the monomials. $$\frac{\left(10 m^{5} n^{4}\right)\left(5 m^{3} n^{6}\right)}{25 m^{7} n^{5}}$$

Divide each polynomial by the monomial. $$\left(63 a^{2} b^{3}+72 a b^{4}\right) \div(9 a b)$$

Divide each polynomial by the monomial. $$\left(16 y^{2}-20 y\right) \div 4 y$$

Divide each polynomial by the monomial. $$\frac{51 m^{4}+72 m^{3}}{-3}$$

Divide each polynomial by the binomial. $$\left(a^{2}-2 a-35\right) \div(a+5)$$
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