Problem 97 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(\frac{3 y^{5}}{6 x^{4}}\right)^{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{y^{15}}{8 x^{12}} \).

Step 1: Apply the Power Rule to the Fraction

The expression \( \left(\frac{3 y^5}{6 x^4}\right)^3 \) can be simplified using the power rule \( (\frac{a}{b})^n = \frac{a^n}{b^n} \). Apply this to the fraction:\[ \left(\frac{3 y^5}{6 x^4}\right)^3 = \frac{(3 y^5)^3}{(6 x^4)^3} \] This separates the power to both numerator and denominator.

Step 2: Simplify the Numerator

Now, simplify the numerator \((3 y^5)^3\) by applying the power rule \((a^m)^n = a^{m \cdot n}\):\[ (3 y^5)^3 = 3^3 \cdot (y^5)^3 = 27 \cdot y^{15} \] This results in \(27 y^{15}\) for the numerator.

Step 3: Simplify the Denominator

Next, simplify the denominator \((6 x^4)^3\) in a similar manner:\[ (6 x^4)^3 = 6^3 \cdot (x^4)^3 = 216 \cdot x^{12} \] This gives us \(216 x^{12}\) for the denominator.

Step 4: Form the New Fraction

Now, replace the simplified numerator and denominator back into the fraction:\[ \frac{27 y^{15}}{216 x^{12}} \]

Step 5: Simplify the Fraction Further

Finally, simplify the fraction by reducing the coefficient and see if further simplification is possible:- The numbers 27 and 216 have a common factor of 27. Divide both by 27: \[ \frac{27}{216} = \frac{1}{8} \]- The variables \(y^{15}\) and \(x^{12}\) remain as they cannot be simplified further relative to each other.Thus, the simplified form of the expression is:\[ \frac{y^{15}}{8 x^{12}} \]

Key Concepts

Power Rule

The power rule is a fundamental tool in algebra which helps simplify expressions involving exponents. This rule states that when you have a power raised to another power, you can multiply the exponents together. For example, when you have \( (a^m)^n \), it simplifies to \( a^{m \cdot n} \).
This becomes particularly useful when dealing with fractions raised to powers. According to the power rule for fractions, \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \).
By applying this rule, every part of the fraction gets its exponent raised to the power respectively, helping to simplify complex algebraic expressions.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator no longer have common factors other than 1.
For example, in the fraction \( \frac{27}{216} \), both numbers are divisible by 27. Dividing the numerator and the denominator by 27 simplifies this to \( \frac{1}{8} \).Some key steps in fraction simplification include:

• Finding the greatest common divisor (GCD) of the numerator and the denominator.
• Dividing both the numerator and the denominator by their GCD.
• Ensuring no common factors remain for the simplest form.

By following these steps, you can simplify any complicated fraction into a more manageable expression.

Exponentiation

Exponentiation involves raising a number or expression to a power, which means multiplying the number by itself a certain number of times. For example, \( x^3 \) means \( x \times x \times x \).
In algebra, you often have to perform exponentiation on expressions, such as raising fractions or entire expressions to a power. This is where the power rule comes into play, making it simpler to handle such operations.
Exponentiation is essential for expanding expressions and simplifying algebraic equations efficiently.

Numerators and Denominators

In any fraction, the numerator is the top number while the denominator is the bottom number. Together, they represent parts of a whole. In the given expression, \( \frac{3 y^5}{6 x^4} \),
the numerator \( 3 y^5 \) and the denominator \( 6 x^4 \) play different roles in the fraction.Here's what to remember about them:

• The numerator divides the fraction into parts.
• The denominator specifies the total possible parts.
• Simplifying each part separately can make the fraction easier to work with.

Being comfortable with handling numerators and denominators is vital for working efficiently with algebraic fractions.