Problem 60 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(\frac{3 x}{5 y}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\left(\frac{3x}{5y}\right)^2$. Answer: $\frac{9x^2}{25y^2}$

Step 1: Apply power rule to the numerator and denominator separately

We can apply the power of 2 to both the numerator (3x) and the denominator (5y) separately: $$ \left(\frac{3x}{5y}\right)^2 = \frac{(3x)^2}{(5y)^2} $$

Step 2: Apply power rule again for each term in the numerator and denominator

Now, we will apply the power of 2 to each part within the numerator and denominator: $$ \frac{(3x)^2}{(5y)^2} = \frac{3^2x^2}{5^2y^2} $$

Step 3: Simplify constants and write the final expression

Finally, we can simplify the constants (3^2 and 5^2) and rewrite the expression in a simplified form: $$ \frac{3^2x^2}{5^2y^2} = \frac{9x^2}{25y^2} $$ The simplified expression is: $$ \left(\frac{3x}{5y}\right)^2 = \frac{9x^2}{25y^2} $$

Key Concepts

Simplifying Expressions

Simplifying expressions means reducing a complex expression into a simpler or more concise form. When we simplify, we aim to make expressions more usable without changing their value. In algebra, this often involves combining like terms or applying specific mathematical rules.

In the given exercise, we started with the expression \(\left(\frac{3x}{5y}\right)^2\). Simplification involved using the power rules for exponents to treat the expression by breaking it down into more manageable parts.

• First, focus on the numerator and the denominator separately.
• After applying the exponent, it resulted in \(\frac{(3x)^2}{(5y)^2}\). This step expands each entity within the fraction.
• Finally, by evaluating and simplifying each component, we ended up with \(\frac{9x^2}{25y^2}\).

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. These expressions often contain variables raised to a power or combined in some way. Handling algebraic fractions requires a careful application of algebraic rules.

In our problem, the algebraic fraction \(\frac{3x}{5y}\) is raised to a power. When dealing with such fraction expressions, we can manage them by:

• Applying exponent rules separately to both the numerator and denominator.
• Maintaining the balance of the fraction by applying the power rule uniformly across both parts.
• Simplifying constants independently, which helps in reducing the fraction to its simplest form.

The final expression \(\frac{9x^2}{25y^2}\) represents the simplified version, where \(3^2\) becomes 9 and \(5^2\) becomes 25.

Exponent Rules

Exponent rules provide us with shortcuts for dealing with mathematical powers. These rules are essential for simplifying expressions, especially when the expressions involve variables.

The power rule, which states that \((a^m)^n = a^{m \cdot n}\), helps us handle situations where expressions have exponents. By applying this rule to both the numerator and the denominator, we manage smaller computations.

• For the numerator \((3x)^2\): Apply the power separately to get \(3^2 \cdot x^2\).
• For the denominator \((5y)^2\): Similarly, apply the power separately, resulting in \(5^2 \cdot y^2\).
• Combine these results to achieve the final simplified form.

Exponent rules like this not only simplify calculations but also reduce the work involved in solving complex algebraic expressions.