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Chapter 5: Problem 80

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{x^{5}}{y^{2}}\right)^{7} $$

Short Answer

Expert verified
\( \frac{x^{35}}{y^{14}} \)

Step by step solution

01

Understand the Power Rule

To simplify expressions that involve exponents, use the power rule: \(\bigg(\frac{a}{b}\bigg)^m = \frac{a^m}{b^m}\). This means each element inside the parentheses is raised to the power outside the parentheses.
02

Apply the Power Rule

Apply the power rule to our expression: \(\bigg(\frac{x^5}{y^2}\bigg)^7 = \frac{(x^5)^7}{(y^2)^7}\). This will allow us to simplify each part of the fraction separately.
03

Simplify the Numerator

Simplify the numerator by using the power of a power rule: \((x^5)^7 = x^{5 \times 7} = x^{35}\).
04

Simplify the Denominator

Next, simplify the denominator using the power of a power rule: \((y^2)^7 = y^{2 \times 7} = y^{14}\).
05

Write the Simplified Fraction

Combine the simplified numerator and denominator to get the final simplified expression: \(\frac{x^{35}}{y^{14}}\).

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Key Concepts

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

Power Rule
The power rule is a fundamental concept in algebra, especially when dealing with exponents. It states that \(\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}\). This means you raise both the numerator and the denominator inside the parentheses to the power indicated outside the parentheses.
For example: If you have \(\left( \frac{x^5}{y^2} \right)^7\), you would apply the power rule to both x and y individually.
  • Step 1: Recognize that each element inside the parentheses is affected.
  • Step 2: Raise the numerator \(x^5\) to the power 7, resulting in \(x^{5 \times 7}\), which simplifies to \(x^{35}\).
  • Step 3: Raise the denominator \(y^2\) to the power 7, resulting in \(y^{2 \times 7}\), which simplifies to \(y^{14}\).
Applying the power rule correctly allows you to simplify complex expressions systematically and accurately.
Exponents
Exponents are a shorthand way to express repeated multiplication of a number by itself. For example, \(x^5\) means \(x\) is multiplied by itself 5 times.
There are few essential rules governing exponents:
  • Product Rule: \(a^m \cdot a^n = a^{m+n}\).
  • Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\).
  • Power of a Power Rule: \( (a^m)^n = a^{m \times n}\).
  • Power of a Product Rule: \( (ab)^m = a^m \cdot b^m\).
  • Power of a Quotient Rule: \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\).
Understanding these rules helps to manipulate and simplify algebraic expressions efficiently. For instance, in our exercise, the power of a power rule is used twice - once for the numerator and once for the denominator.
\( (x^5)^7 = x^{35}\) and \( (y^2)^7 = y^{14}\). These steps are crucial for reaching the simplified form of the original problem.
Fractions in Algebra
Fractions often appear in algebraic expressions, and knowing how to simplify them is crucial. A fraction in algebra consists of a numerator (top part) and a denominator (bottom part).
To simplify a fraction:
  • Identify common factors in the numerator and the denominator.
  • Use the power rule and other exponent rules to break down complex expressions.
In the problem: \(\left(\frac{x^5}{y^2}\right)^7\), you treat both the numerator \(x^5\) and the denominator \(y^2\) independently and apply the power rule to each.
Raising \(x^5\) to the power 7 gives \(x^{35}\), and raising \(y^2\) to the power 7 gives \(y^{14}\).
Finally, combine these results to get the simplified fraction: \(\frac{x^{35}}{y^{14}}\).
These steps ensure the fraction is fully simplified by correctly applying algebraic rules.

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Most popular questions from this chapter

Simplify. \(\frac{9}{2} x^{8}+\frac{1}{9} x^{2}+\frac{1}{2} x^{9}+\frac{9}{2} x+\frac{9}{2} x^{9}+\frac{8}{9} x^{2}+\frac{1}{2} x-\frac{1}{2} x^{8}\)

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{x^{3}}{y^{2} z}\right)^{5} $$

Find the domain of \(f / g,\) if $$ f(x)=\frac{3 x}{2 x+5} \quad \text { and } \quad g(x)=\frac{x^{4}-1}{3 x+9} $$

Compare \(8 \times 10^{-90}\) and \(9 \times 10^{-91} .\) Which is the larger value? How much larger? Write scientific notation for the difference.

Simplify. \(\frac{1}{2} x-\frac{1}{4}-\left(\frac{3}{2} x+\frac{3}{4}\right)\)
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