Problem 57 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(\frac{9 x^{4}}{y^{10}}\).

Step 2: 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 \(\left(\frac{9 x^{4}}{y^{10}} \right) \cdot \left(\frac{x}{y} \right ) = \frac{9 x^{4} \cdot x}{y^{10} \cdot y} \ =\ \frac{9 x^{5}}{y^{11}} \).

Key Concepts

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.