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Chapter 5: Problem 57

\(\left(\frac{3 x^{2}}{y^{5}}\right)^{2}\left(\frac{x}{y}\right)\)

Short Answer

Expert verified
\frac{9 x^{5}}{y^{11}}

Step by step solution

01

Apply the power of a power rule

Use the rule \(a^m\) ^n = a^{m \times n} to simplify the first part of the expression. \(\left(\frac{3 x^{2}}{y^{5}}\right)^{2} = \frac{3^2 (x^2)^2}{(y^5)^2}\). This simplifies to \(\frac{9 x^{4}}{y^{10}}\).
02

Multiply the fractions

Multiply the simplified expression by the second fraction \(\frac{x}{y}\), using the rule \ \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\. This gives \(\left(\frac{9 x^{4}}{y^{10}} \right) \cdot \left(\frac{x}{y} \right ) = \frac{9 x^{4} \cdot x}{y^{10} \cdot y} \ =\ \frac{9 x^{5}}{y^{11}} \).

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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
The power of a power rule is a fundamental principle in algebra. This rule states that when you raise a power to another power, you multiply the exponents. For example, if you have \( (a^m)^n \), it simplifies to \( a^{m \times n} \).

In our exercise, we applied this rule to \( \left( \frac{3 x^{2}}{y^{5}} \right)^{2} \). By multiplying the exponents, this becomes \( \frac{3^2 (x^2)^2}{(y^5)^2} = \frac{9 x^{4}}{y^{10}} \).

Remember to carefully apply the power to every part of the fraction. This ensures that both the numerator and the denominator are correctly simplified.
multiplying fractions
Multiplying fractions is another key concept in algebra. You can multiply fractions by multiplying their numerators together and their denominators together.

The formula \( \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \) demonstrates this. Multiplying \( \frac{9 x^{4}}{y^{10}} \) by \( \frac{x}{y} \) involves multiplying the numerators \( (9 x^{4} \cdot x) \) and the denominators \( (y^{10} \cdot y) \), giving us \( \frac{9 x^{5}}{y^{11}} \).

Make sure to multiply both parts of each fraction and then simplify if possible.
exponents
Exponents are used to indicate how many times a number is multiplied by itself. For example, in \( x^5 \), the number \( x \) is multiplied by itself five times.

When dealing with exponents in fractions, such as in \( \frac{3 x^{2}}{y^{5}} \), each part of the fraction must be considered separately.

In our exercise, simplifying \( \left( \frac{3 x^{2}}{y^{5}} \right)^{2} \) required us to apply the exponents to the numerator, \( x^2 \), and the denominator, \( y^5 \), individually, resulting in \( \frac{3^2 (x^2)^2}{(y^5)^2} = \frac{9 x^{4}}{y^{10}} \).

Understanding how to work with exponents is crucial for simplifying algebraic expressions accurately.

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Most popular questions from this chapter

\(\left(3 x^{2}+19 x+5\right) \div(x+1)\)

Problem: Simplify: \((2 p-9)\left(3 p^{2}+6 p-5\right)\) Incorrect Answer: $$ \begin{aligned} &(2 p-9)\left(3 p^{2}+6 p-5\right) \\ &=2 p\left(3 p^{2}\right)+2 p(6 p)+2 p(-5)-9\left(3 p^{2}\right)-9(6 p)-9(-5) \\\ &=6 p^{3}+12 p^{2}+10 p-27 p^{2}-54 p+45 \\ &=6 p^{3}-15 p^{2}-44 p+45 \end{aligned} $$

\(\left(2 x^{2}+9 x+20\right) \div(x+4)\)

\(\left(x^{2}-25\right) \div(x+5)\)

\(\left(5 w^{4}-15 w^{2}+60 w+20\right) \div(-5 w)\)
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