Question

Simplify the expression.\(\left(27 x^{6} y^{9}\right)^{2 / 3}\)

Short answer

The simplified form of the expression is \(9* x^{4} * y^{6}\)

Step-by-step solution

Identify the expression

The given expression to simplify is \((27 x^{6} y^{9})^{2 / 3}\)

Apply the rule of exponents

We can apply the rule of exponents, which states that when you have a power of a power, you multiply the powers. So, \((27 x^{6} y^{9})^{2 / 3}\) simplifies to \(27^{2/3} * x^{(6*(2/3))} * y^{(9*(2/3))}\)

Simplify further

Now, simplify \(27^{2/3}\), \(x^{6*(2/3)}\) and \(y^{9*(2/3)}\). This results in \(9* x^{4} * y^{6}\)

Key concepts

Power of a Power Rule

When dealing with exponents, one important rule is the power of a power rule. This states that

• For any base raised to an exponent that is itself raised to another exponent, you multiply the exponents together.
• Mathematically, this is written as \[ (a^m)^n = a^{m\cdot n} \].

In the exercise we're considering, each component of the expression \((27 x^6 y^9)^{2/3}\) is raised to the power of \(2/3\). According to the power of a power rule, you'll handle 27, \(x^6\), and \(y^9\) individually.

• For 27, you'll address it as \(27^{2/3}\).
• For \(x^6\), multiply the existing exponent by \(2/3\), leading to \(x^{6 \cdot (2/3)} = x^4\).
• Similarly, for \(y^9\), you'll get \(y^{9 \cdot (2/3)} = y^6\).

Understanding this rule makes managing complex expressions easier.

Algebraic Simplification

Algebraic simplification is about making expressions easier to work with. This involves breaking down complex equations by following established rules like combining like terms and simplifying radicals.

When simplifying the expression \((27 x^6 y^9)^{2 / 3}\), we need to simplify each component separately. After applying the power of a power rule:

• The expression \(27^{2/3}\) involves finding a simpler equivalent of raising a number to a fractional exponent.
• The exponents for \(x\) and \(y\) terms are fractions but simplify to whole numbers: \(x^4\) and \(y^6\), respectively.

For \(27^{2/3}\), it's telling us to find the cube root of 27 first, which is 3, and then raise it to the power of 2, resulting in 9. Hence, the complete solution is \(9x^4y^6\). Simplification thus involves several steps of systematic reduction.

Rational Exponents

Rational exponents refer to exponents that are fractions. These are particularly useful as they relate to roots, offering a compact notation for radicals. For example,

• \(a^{1/n}\) signifies the \(n\)-th root of \(a\).
• \(a^{m/n}\) therefore means taking the \(n\)-th root first, and then raising the result to the power of \(m\).

In our example \((27 x^6 y^9)^{2/3}\), you see this concept in action:

• The \(27^{2/3}\) simplifies by first taking the cube root of 27 (yielding 3), and then squaring it (resulting in 9).

Dealing with rational exponents allows us to transition seamlessly between dealing with roots and powers, providing a powerful tool for algebraic simplification. Practicing these concepts with different exercises boosts clarity and fluency in mathematical expressions.